# What are Capra-Convex Sets?

Adrien Le Franc, Jean-Philippe Chancelier, Michel de Lara, Seta Rakotomandimby

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Adrien Le Franc, Jean-Philippe Chancelier, Michel de Lara, Seta Rakotomandimby. What are Capra-Convex Sets?. 2025. hal-05243799

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# What are Capra-Convex Sets?

Adrien Le Franc*

Jean-Philippe Chancelier*

Michel De Lara*

Seta Rakotomandimby*

September 7, 2025

## Abstract

This paper focuses on a specific form of abstract convexity known as Capra-convexity, where a constant along primal rays (Capra) coupling replaces the scalar product used in standard convex analysis to define generalized Fenchel conjugacies. A key motivating result is that the $l_0$ pseudonorm — which counts the number of nonzero components in a vector — is equal to its Capra-biconjugate. This implies that $l_0$ is a Capra-convex function, highlighting potential applications in statistics and machine learning, particularly for enforcing sparsity in models. Building on prior work characterizing the Capra-subdifferential of $l_0$ and the role of source norms in defining the Capra-coupling, the paper provides a characterization of Capra-convex sets.

**Keywords** Generalized subdifferential; $l_0$ pseudonorm; Sparsity; Capra-coupling

## 1 Introduction

From an historical perspective, convexity was first studied as a geometrical property of sets [6]. The formal contemporary definition of a convex function was seemingly introduced by Jensen [10], in the context of a growing interest for such functions at the dawn of the XIXth century. The development of modern convex analysis, with the prominent role of convex conjugacies, is mostly due to Fenchel, Moreau and Rockafellar over the XXth century, as exposed in the bibliographical notes of [9].

Due to the success of convexity in solving optimization problems, there has been several attempts to extend this theory to larger classes of sets, functions and conjugacies. For instance, starting from the geometrical roots of convexity, spherical convexity was introduced for the study of sets contained in the Euclidean sphere: the definition of a spherically-convex set follows the one of a classical convex set, in the sense that the geodesic between two points of a spherical set must remain inside the set [7]. Conversely, other extensions of convexity originate from abstract conjugacies, which derive from the choice of a general set of functions

<footnote>*CERMICS, École nationale des ponts et chaussées, IP Paris, France</footnote>
to play the role of affine minors in the classical definition of the Fenchel conjugate and biconjugate [17].

In this paper, we focus on a specific type of abstract convexity, named Capra-convexity, where the so-called Capra (constant along primal rays) coupling function plays the role of the scalar product in usual convexity, to generalize the set of affine minors in usual Fenchel conjugacies. A principal result of Capra-convexity, which motivates our interest, is that the $l_0$ pseudonorm — which counts the nonzero components of a vector — is equal to its Capra-biconjugate, and is therefore a Capra-convex function [2]. This suggests potential applications to statistics and machine learning, where the $l_0$ pseudonorm is extensively used to enforce sparsity in statistical models. From the perspective of convex analysis, further investigations have allowed to better understand the role of a source norm in the definition of the Capra-coupling [5] and to characterize the Capra-subdifferential of $l_0$ [11]. Moreover, algorithmic approaches based on a generalized cutting plane method have been investigated recently in [13]. In addition to these previous developments, and in the perspective of minimizing $l_0$ — or any Capra-convex function of interest — over a constraint set which somehow preserves Capra-convexity, we now ask:

what are Capra-convex sets?

In order to address this question, we first introduce background results on usual convexity and Capra-convexity hereafter in §1. Second, we propose a definition for Capra-convex sets in §2 and, as our main results, we provide explicit characterizations of such sets. Subsequently, we investigate on the relationship between Capra-convex sets and conical hulls, which allows a connection between closed spherically-convex and Capra-convex sets. Third, we propose a collection of examples in §3 to illustrate our main results. To ease the reading of the paper, most technical proofs are given in Appendix A.

## 1.1 Convex functions and sets

We introduce some basic notions and refer to [15, 1] for more advanced materials.

We denote by $\mathbb{R} = \mathbb{R} \cup \{+\infty, -\infty\}$ the extended real line. We consider the Euclidean space $\mathbb{R}^n$, equipped with the scalar product $\langle \cdot | \cdot \rangle$. For any function $h: \mathbb{R}^n \to \mathbb{R}$, its epigraph is the set $\text{epih} = \{(w, t) \in \mathbb{R}^n \times \mathbb{R} | h(w) \le t\}$. We recall that the function $h$ is convex if and only if its epigraph is a convex set. For any set $X \subseteq \mathbb{R}^n$, $\iota_X: \mathbb{R}^n \to \mathbb{R}$ and $\sigma_X: \mathbb{R}^n \to \mathbb{R}$ denote respectively the indicator function and the support function of the set $X$, defined as

$$ \iota_X(x) = 0 \text{ if } x \in X, \quad \iota_X(x) = +\infty \text{ if } x \notin X. \quad (1a) $$

$$ \sigma_X(y) = \sup_{x \in X} \langle x | y \rangle, \quad \forall y \in \mathbb{R}^n. \quad (1b) $$

We denote by $\text{co}(X)$ the convex hull of the set $X$ (defined as the smallest convex subset of $\mathbb{R}^n$ containing $X$), and by $\overline{\text{co}}(X)$ the closed convex hull of $X$ (defined as the closure of $\text{co}(X)$).

We pay a special attention to cones, which are sets $K \subseteq \mathbb{R}^n$ such that $\lambda K \subset K$ for all $\lambda > 0$. Notice that, with this definition, a cone does not necessarily contain the origin, and
is not necessarily convex, as will be illustrated with Capra-convex sets in §2. We say that a cone $K$ is pointed if $K \cap (-K) \subseteq \{0\}$. A cone can be constructed from any subset $X \subseteq \mathbb{R}^n$, by means of the conical hull of $X$, defined in [1, Definition 6.1] as

$$ \mathrm{cone}(X) = \{\lambda x \mid x \in X, \lambda > 0\}, \qquad (2a) $$

or by means of the positive hull of $X$, defined in [15, Chapter 3, §G] as

$$ \mathrm{pos}(X) = \{\lambda x \mid x \in X, \lambda \ge 0\} = \mathrm{cone}(X) \cup \{0\}. \qquad (2b) $$

We stress that this definition of the conical hull omits the value $\lambda = 0$, so that if $0 \notin X$ then $0 \notin \mathrm{cone}(X)$. By contrast, $0 \in \mathrm{pos}(X)$. We also respectively denote by $\mathrm{cone}(X)$ and $\mathrm{pos}(X)$ the topological closures of the conical hull and the positive hull of a set.

## 1.2 Basic notions in Capra-convexity

We start by recalling the definition of the Capra coupling. In what follows, let $\|\cdot\|$ be a norm on $\mathbb{R}^n$, referred to as the source norm. We respectively denote $S = \{x \in \mathbb{R}^d \mid \|x\| = 1\}$ and $B = \{x \in \mathbb{R}^d \mid \|x\| \le 1\}$ the unit sphere and unit ball associated with the source norm.

**Definition 1.1** We define the set

$$ S^{(0)} = S \cup \{0\}, \qquad (3) $$

and the radial projection

$$ \varrho: \mathbb{R}^n \to \mathbb{R}^n, \quad \varrho(x) = \frac{x}{\|x\|} \quad \text{if } x \neq 0, \quad \varrho(x) = 0 \quad \text{if } x = 0. \qquad (4) $$

We will use the following easy-to-establish properties:

$$ \varrho(\mathbb{R}^n) = S^{(0)}, \qquad (5a) $$

$$ (\text{zero-homogeneity}) \quad \varrho(\lambda x) = \varrho(x), \quad \forall \lambda \in \mathbb{R} \setminus \{0\}, \quad x \in \mathbb{R}^n, \qquad (5b) $$

$$ \text{if } K \subset \mathbb{R}^n \text{ is a cone, then } \varrho(K) = K \cap S^{(0)}, \qquad (5c) $$

$$ \varrho^{-1}(X) = \varrho^{-1}(X \cap S^{(0)}) = \mathrm{cone}(X \cap S^{(0)}), \quad X \subset \mathbb{R}^n, \qquad (5d) $$

**Definition 1.2** ([3], **Definition 4.1**) We define the Capra coupling $\phi: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ between $\mathbb{R}^n$ and $\mathbb{R}^n$, by

$$ \forall y \in \mathbb{R}^n, \quad \phi(x, y) = \langle \varrho(x) | y \rangle = \begin{cases} \frac{\langle x | y \rangle}{\|x\|}, & \text{if } x \neq 0, \\ 0, & \text{if } x = 0. \end{cases} \qquad (6) $$
A coupling function such as the Capra coupling $\zeta$ given in Definition 1.2 gives rise to generalized Fenchel-Moreau conjugacies [17, 12], that we briefly recall. Let us consider a function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$. The $\zeta$-Fenchel-Moreau conjugate of $f$ is the function $f^\zeta: \mathbb{R}^n \to \overline{\mathbb{R}}$ defined by

$$f^\zeta(y) = \sup_{x \in \mathbb{R}^n} (\zeta(x, y) - f(x)), \quad \forall y \in \mathbb{R}^n, \qquad (7a)$$

the $\zeta$-Fenchel-Moreau biconjugate of $f$ is the function $f^{\zeta\zeta'}: \mathbb{R}^n \to \overline{\mathbb{R}}$ defined by

$$f^{\zeta\zeta'}(x) = \sup_{y \in \mathbb{R}^n} (\zeta(x, y) - f^\zeta(y)), \quad \forall x \in \mathbb{R}^n, \qquad (7b)$$

and we have the inequality

$$f^{\zeta\zeta'}(x) \leq f(x), \quad \forall x \in \mathbb{R}^n. \qquad (7c)$$

Observe that, if we replace the Capra coupling $\zeta$ with the scalar product $\langle \cdot | \cdot \rangle$ in (7), we retrieve the well-known notions of Fenchel conjugate $f^*$ and biconjugate $f^{**'}$ in standard convex analysis. We refer to [3] for a more complete introduction to Capra conjugacies.

**Definition 1.3** We say that the function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ is Capra-convex iff we have an equality in (7c), that is, if $f^{\zeta\zeta'} = f$.

We define the support of a vector $x = \{x_j\}_{j \in [1, n]} \in \mathbb{R}^n$ by $\text{supp}(x) = \{j \in [1, n] | x_j \neq 0\}$. The $\ell_0$ pseudonorm is the function $\ell_0: \mathbb{R}^n \to [1, n]$ defined by

$$\ell_0(x) = |\text{supp}(x)|, \quad \forall x \in \mathbb{R}^n, \qquad (8)$$

where $|K|$ denotes the cardinality of a subset $K \subseteq [1, n]$. We recall a central result which motivates our interest for Capra-convexity: for a certain class of source norms, which encom-passes e.g. the $\ell_p$ norms for $1 < p < +\infty$, the $\ell_0$ pseudonorm is a Capra-convex functions, in the sense of Definition 1.3. We refer to [4] for more details on the choice of a suitable source norm to enforce the Capra-convexity of $\ell_0$.

# 2 Capra-convex sets

In §2.1, we define Capra-convex sets and present our main results regarding the characterization of such sets. Then, in §2.2, we present sufficient conditions for the conical hull of a set to be Capra-convex, and we discuss of the links between Capra-convex and spherically-convex sets. To ease the reading of this section, the proofs of the main results are relegated to Appendix A.

## 2.1 Definition and characterization of Capra-convex sets

We start with the definition of Capra-convex sets. Subsequently, we outline our main results regarding the characterization of such sets.
**Definition of Capra-convex sets using indicator functions.** Closed convex sets play a central role in convex analysis: for instance, proper functions with closed convex epigraphs are stable by the Fenchel-Moreau biconjugate. In particular, for a closed convex set $X \subseteq \mathbb{R}^n$, its indicator function $\iota_X$ is convex l.s.c., so that $\iota_X = \iota_X^{\star\star'}$, even for the — unique — nonproper case where $X = \emptyset$, as $\iota_\emptyset = \iota_\emptyset^{\star\star'} = +\infty$. By analogy with closed convex sets in classical convex analysis, we provide the following definition of Capra-convex sets.

**Definition 2.1** Let $\|\cdot\|$ be a source norm, and $\cdot$ be the corresponding Capra-coupling as in Definition 1.2. We say that the set $X \subseteq \mathbb{R}^n$ is Capra-convex if the indicator function $\iota_X$ is a Capra-convex function.

With this definition, we deduce the following immediate property.

**Proposition 2.2** If a set $X \subseteq \mathbb{R}^n$ is Capra-convex, then $X$ is a cone.

**Proof.** The coupling $\cdot$ in Definition 1.2 is one-sided linear (see [2, Definition 2.3]) and factorizes with the radial projection $\rho: \mathbb{R}^n \to S^{(0)}$ in (4), so that $\iota_X = \iota_X^{\star\star'} = \iota_X^{\star\star'} \circ \rho$, see [2, Proposition 2.5]. Thus, for any $\lambda > 0$, one has that

$$ \iota_{\lambda X}(x) = \iota_X(x/\lambda) = \iota_X^{\star\star'}(\rho(x/\lambda)) = \iota_X^{\star\star'}(\rho(x)) = \iota_X(x), \quad \forall x \in \mathbb{R}^n, $$

where we have used the property that the radial projection $\rho$ is zero-homogeneous. We conclude that $\lambda X = X$ for any $\lambda > 0$, which proves that $X$ is a cone. $\square$

**Main results on the characterization of Capra-convex sets.** We will see that a Capra-convex set needs not be convex, and that, somehow more surprisingly, it needs not be closed either. Our principal result is the following generic characterization of Capra-convex sets.

**Theorem 2.3** Let $\|\cdot\|$ be a source norm on $\mathbb{R}^n$, $\rho: \mathbb{R}^n \to S^{(0)}$ be the radial projection defined in (4), and $\cdot$ be the corresponding Capra-coupling, as in Definition 1.2. Let $K \subseteq \mathbb{R}^n$ be a set. We have that

$$ K \text{ is Capra-convex} \iff K \text{ is a cone and } \rho(K) = \overline{\text{co}}(\rho(K)) \cap S^{(0)}. \quad (9) $$

To illustrate Theorem 2.3, we provide an example of Capra-convex set in Figure 1 which satisfies the characterization (9). Moreover, to give a more geometrical interpretation of (9), we recall (5c): if the set $K$ is a cone, then $\rho(K) = K \cap S^{(0)}$. It is also insightful to observe that the equality between sets in (9) is equivalent to $K = \rho^{-1}(\overline{\text{co}}(\rho(K)))$, so that the set $\rho^{-1}(\overline{\text{co}}(\rho(K)))$ can be interpreted as the Capra-convex hull of $K$. Nevertheless, the characterization of (9) remains quite formal, and we now provide more practical conditions to identify Capra-convex sets.
**Corollary 2.4** Under the hypotheses of Theorem 2.3, we have that

$$ K \text{ is Capra-convex} \implies \begin{cases} K \text{ is a cone,} \\ K \cup \{0\} \text{ is closed,} \\ K \cap \{0\} = \overline{\mathbb{C}}(\varrho(K)) \cap \{0\}. \end{cases} \qquad (10) $$

Referring again to Figure 1, we observe that, for the example of set K in Figure 1a, we have indeed that $K \cap \{0\} = \overline{\mathbb{C}}(\varrho(K)) \cap \{0\} = \emptyset$. Thus, in practice, checking whether (10) holds often boils down to checking whether $0 \in \overline{\mathbb{C}}(\varrho(K))$ holds, which can be a convenient alternative to (9) — we refer to the examples of §3.1.

One may naturally wonder whether the reverse implication holds in (10). Interestingly, the answer to that question depends on the geometry of the unit ball induced by the source norm. Indeed, we will show a sufficient condition to have the equivalence in (10) which depends on the following property of balls. We recall that the unit ball B of a norm $\|\cdot\|$ is rotund when the corresponding sphere S coincides with the extreme points of B. For instance, this is the case of the $l_p$ norms for values of p satisfying $1 < p < \infty$.

**Corollary 2.5 (rotund norm balls)** Under the hypotheses of Theorem 2.3, suppose more-over that the unit ball of the source norm $\|\cdot\|$ is rotund. Then, the set K is Capra-convex if and only if the three conditions in the right-hand side of (10) are satisfied.

When the unit ball of the source norm $\|\cdot\|$ is not rotund, the reverse implication in (10) might fail, as we illustrate with an example in §3.1. Lastly, in addition to Corollary 2.5, and regardless of the source norm $\|\cdot\|$, we identify the following notable cases of Capra-convex sets.

**Corollary 2.6 (closed convex cones)** Let $K \subseteq \mathbb{R}^n$ be a closed convex cone. We have that

(i) the set K is Capra-convex,

(ii) if, moreover, the cone K is pointed, then the set $K \setminus \{0\}$ is Capra-convex.

## 2.2 Capra-convex conical hulls and relationship with spherical convexity

First, we discuss in §2.2.1 the role of Capra-convex conical hulls in optimization problems.

Second, in §2.2.2 we give sufficient conditions for a set to have a Capra-convex conical hull.

Finally, we end in §2.2.3 with a comparison between Capra-convex sets and spherically-convex sets.
<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of Theorem 2.3. Example of a set K for which $\varrho(K)$ (in red in Figure 1a) is equal to $\overline{\text{co}}(\varrho(K)) \cap S^{(0)}$ (in red in Figure 1b) when the source norm is $\|\cdot\|_{\infty}$</figcaption></figure>

### 2.2.1 Closed convex factorization of Capra-convex problems

Capra-convex constrained minimization problems enjoy a closed convex factorization property. More specifically, let us consider the optimization problem

$$ \inf_{x \in X} f(x), \qquad (11a) $$

defined after an objective function $f: \mathbb{R}^n \to \mathbb{R}$ and a constraint set $X \subseteq \mathbb{R}^n$. If both the objective function $f$ and the indicator function $\iota_X$ are Capra-convex, then we recall, according to [2, Proposition 2.6], that the function $f + \iota_X$ factorizes as $f + \iota_X = \mathcal{F} \circ \varrho$, where $\mathcal{F}: \mathbb{R}^n \to \overline{\mathbb{R}}$ is a proper l.s.c. convex function, so that Equation (11a) becomes

$$ \inf_{x \in X} f(x) = \inf_{x \in \mathbb{R}^n} \mathcal{F} \circ \varrho(x) = \inf_{s \in S^{(0)}} \mathcal{F}(s) = \inf(\mathcal{F}(0), \inf_{s \in S} \mathcal{F}(s)). \qquad (11b) $$

In such a case, solving the optimization Problem (11a) amounts to minimizing the closed convex [14, p. 15] function $\mathcal{F}$ over the unit sphere $S$.

However, minimization problems involving Capra-convex objective functions are usually not defined on a Capra-convex set of constraints. For instance, among minimal cardinality problems, i.e. instances with $f = \ell_0$ in Problem (11a), the fundamental sparse problem in compressive sensing has a constraint set of type $X = \{x \in \mathbb{R}^n \mid Ax = b\}$ (see e.g. [16]). Yet, when $b \neq 0$, the resulting affine space is not a cone, so that $X$ cannot be a Capra-convex set according to Proposition 2.2.

Nonetheless, Capra-convex functions $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ are 0-homogeneous, that $f(\lambda x) = f(x)$, for any $x \in \mathbb{R}^n$ and any real number $\lambda > 0$. Thus, for such functions, the problem (11a) can be rewritten as

$$ \inf_{u \in \text{cone}(X)} f(u), \qquad (12) $$
using the change of variable $x = \lambda u, u \in \mathbb{R}^n, \lambda > 0$. Therefore, even if the set of constraints $X$ is not Capra-convex, its conical hull $\text{cone}(X)$ might be Capra-convex, under some assumptions that we now investigate on.

### 2.2.2 Sufficient conditions for a set to have a Capra-convex conical hull

We now provide conditions for a set $X \subseteq \mathbb{R}^n$ to have a Capra-convex conical hull. We restrict our interest to cases where $0 \notin X$, which correspond to nontrivial instances of minimal cardinality problems, where $f = l_0$ in Problem (11a).

**Proposition 2.7** Let $X \subseteq \mathbb{R}^n$ be a compact set such that $0 \notin \text{co}(X)$. Let $\|\cdot\|$ be a source norm on $\mathbb{R}^n$ and $\subset$ be the corresponding Capra-coupling as in Definition 1.2.

If one of the following conditions is satisfied:

(i) the unit ball of the source norm $\|\cdot\|$ is rotund;

(ii) the set $X$ is convex;

then $\text{cone}(X)$ is Capra-convex.

We introduce two examples to emphasize that, without the two key assumptions of Proposition 2.7, the conical hull of a set $X \subseteq \mathbb{R}^n$ might fail to be Capra-convex.

We start with an example where $0 \in \text{co}(X)$.

**Example 2.8** Let the set $X \subset \mathbb{R}^2$ be the ball of center $(1, 0)$ and radius $1$. This set is closed, convex and bounded with $0 \in X$. Its conical hull is $\text{cone}(X) = \{x \in \mathbb{R}^2 | x_1 > 0\} \cup \{0\}$, which is not Capra-convex from Corollary 2.5, as $\text{cone}(X) \cup \{0\}$ is not closed (consider for instance the sequence $\{(1/k, 1)\}_{k \ge 1} \subset \text{cone}(X) \cup \{0\}$).

Similarly, we give an example to illustrate that the conical hull $\text{cone}(X)$ might fail to be Capra-convex, if the set $X$ is not compact. In the following example, we get back to the fundamental sparse minimization problem from compressive sensing [16].

**Example 2.9 (Figure 2a)** Let the set $X$ in Problem (11a) be the affine space $H$ of solutions of the linear system $Ax = b$, with a nonzero matrix $A \in \mathbb{R}^{m \times n}$ and a nonzero vector $b \in \mathbb{R}^m$. If $\text{ker}A \neq \{0\}$, then $\text{cone}(X)$ is not Capra-convex.

**Proof.** Let $x \in \text{ker}A$ be nonzero. Let $\bar{x} \in X$. Let $\{\varepsilon_k\}_{k \in \mathbb{N}}$ be a positive sequence converging to 0. For $k \in \mathbb{N}$, we define $x_k = x + \varepsilon_k \bar{x}$ and $x'_k = -x + \varepsilon_k \bar{x}$. As $A(\frac{x_k}{\varepsilon_k}) = A(\frac{x}{\varepsilon_k}) + A(\bar{x}) = 0 + b = b = A(\frac{x_k}{\varepsilon_k})$, we get that $\{x_k, x'_k\} \subset \text{cone}(X)$. It follows that $\{\varrho(x_k), \varrho(x'_k)\} \subset \text{cone}(X) \cap S^{(0)}$ as $\varrho(\mathbb{R}^n) = S^{(0)}$. Now, we observe that

$$ \begin{gathered} \frac{1}{2}\varrho(x_k) + \frac{1}{2}\varrho(x'_k) = \frac{1}{2}\left(\frac{x + \varepsilon_k \bar{x}}{\|x + \varepsilon_k \bar{x}\|} - \frac{x - \varepsilon_k \bar{x}}{\|x - \varepsilon_k \bar{x}\|}\right) \\ \text{(by (4), as } x_k = x + \varepsilon_k \bar{x} \neq 0 \text{ since } A(\frac{x_k}{\varepsilon_k}) = b \neq 0, \text{ and the same for } x - \varepsilon_k \bar{x}) \\ \rightarrow_{k \to +\infty} \frac{1}{2}\left(\frac{x}{\|x\|} - \frac{x}{\|x\|}\right) = 0. \end{gathered} $$
We deduce that $0 \in \overline{\text{co}}(\text{cone}(X) \cap S^{(0)})$. However, $0 \notin \text{cone}(X)$, as $A(0) = 0 \neq b$ hence $0 \notin X$. Thus, using (5c), we obtain that the third condition in the right-hand-side of (10) is not satisfied by $\text{cone}(X)$. We conclude that the set $\text{cone}(X)$ is not Capra-convex, from Corollary (2.4). $\square$

Lastly, regarding Example 2.9, we observe that if we work with a bounded subset $X \subset H$ of the affine space $H$, we retrieve the property that $\text{cone}(X)$ is Capra-convex, from Proposition 2.7. This case is illustrated in Figure 2b.

<figure><img src="image_3.png" /><figcaption>(a) Affine space $H$ in black and $\text{cone}(H)$ in blue (b) Bounded subset $X$ in red and $\text{cone}(X)$ in blue</figcaption></figure>

<caption>Figure 2: Illustration of Example 2.9, where $\text{cone}(H)$ in Fig. 2a (left) is not Capra-convex, but $\text{cone}(X)$ in Fig. 2b (right) is Capra-convex</caption>

### 2.2.3 Relationship with spherical convexity

On the one hand, spherically-convex sets are defined using the unit sphere $S$ and the associated radial projection $\varrho: \mathbb{R}^n \to S^{(0)}$ [7, 8]. Indeed, following [8, Definition 2.5]$^1$ spherically-convex sets are subsets $X \subset S$ of the unit sphere $S$ that are defined by

$$X \text{ is spherically-convex if } \varrho(\lambda x + (1 - \lambda)x') \in X, \quad \forall x, x' \in X, \quad \forall \lambda \in [0, 1]. \quad (13)$$

On the other hand, the unit sphere $S$ and the associated radial projection $\varrho: \mathbb{R}^n \to S^{(0)}$ also appear in the characterization (9) of Capra-convex sets in Theorem 2.3. Hence, it begs the question: is there a link between Capra-convex sets and spherically-convex sets?

First, Capra-convex sets are necessarily cones in $\mathbb{R}^n$ while spherically-convex sets are subsets of the unit sphere $S$. Thus, to make a relevant comparison, we will consider Capra-convex sets and conical hulls in $\mathbb{R}^n$ of spherically-convex sets. We answer the following questions.

<footnote>$^1$We use the definition of spherically-convex from [8] where more general spheres are considered than in [7] where the focus is on the Euclidean sphere. The equivalence of the definitions for the Euclidean sphere is proved by the characterizations [7, Proposition 2] and [8, Proposition 2.7 (iv)].</footnote>
1. Is the conical hull of a spherically-convex set a Capra-convex set? Not necessarily. On the one hand, a spherically-convex set $X$ is not necessarily closed, according to the characterization [8, Proposition 2.7 (iv)], so $\text{cone}(X) \cup \{0\}$ is not necessarily closed. On the other hand, a Capra-convex set $K$ is such that $K \cup \{0\}$ is closed, according to Corollary 2.5.

2. Is the topological closure of the conical hull of a spherically-convex set a Capra-convex set? Yes, see Proposition 2.10.

3. Is the intersection of a Capra-convex set with the unit sphere a spherically-convex set? Not necessarily. See Figure 3c which gives a counterexample of a Capra-convex set $K$ which is not the conical hull of some spherically-convex set. Indeed, the set $K$ is not convex, while conical hulls of spherically-convex sets are convex, according to [7, Proposition 2].

**Proposition 2.10** If the set $X \subset S$ is spherically-convex, then $\overline{\text{cone}}(X)$ is Capra-convex. Moreover, if $\overline{\text{cone}}(X)$ is pointed then $\overline{\text{cone}}(X)\setminus\{0\}$ is Capra-convex.

**Proof.** Using the characterization [8, Proposition 2.7 (iv)], if $X$ is spherically-convex then its positive hull $\text{pos}(X) \subset \mathbb{R}^n$ is convex and pointed. We therefore have that $\overline{\text{pos}}(X)$ is a closed convex cone and as $\overline{\text{pos}}(X) = \overline{\text{cone}}(X)$ that $\overline{\text{pos}}(X)$ is Capra-convex by Corollary 2.6, Item (i). Moreover if $\overline{\text{cone}}(X)$ remains pointed we obtain that $\overline{\text{cone}}(X)\setminus\{0\}$ is Capra-convex by Corollary 2.6, Item (i). $\square$

# 3 Examples of Capra-convex sets

In this section, we showcase examples of Capra-convex sets. We start with generic examples of Capra-convex sets for the $l_p$ source norms in §3.1, and then discuss the properties of the sublevel sets and the epigraph of Capra-convex functions in §3.2.

## 3.1 Examples with the $l_p$ source norms

We now consider the source norms $||\cdot|| = ||\cdot||_p$ with $p \in [1, \infty]$. Our interest in this type of norms is based on a thorough analysis of the Capra-convexity of $l_0$ and of the expression of its Capra-subdifferential for all values of $p \in [1, \infty]$, as exposed in [11]. In particular, we concentrate on the cases $p = 2$, for which the unit ball $B_2$ is rotund, and on the case $p = \infty$, for which the unit ball $B_\infty$ fails to be rotund.

To illustrate the characterizations obtained in §2.1, we provide examples of cones in $\mathbb{R}^2$ and comment on their Capra-convexity. Bearing in mind potential applications to sparse optimization, we are specifically interested in cones $K \subset \mathbb{R}^n$ such that $0 \notin K$. Indeed, when minimizing the $l_0$ pseudonorm on a set or, equivalently, on the conical hull of this set, we
are not interested in trivial cases where the minimum is attained at the origin. Thus, we concentrate on the following examples:

$$ K_1 = \text{cone}(\{(1,0), (-1,1), (-1,-1)\}) \quad (\text{Figure 3a}), \quad (14a) $$

$$ K_2 = \text{cone}(\{(-1,0), (-1,1), (-1,-1)\}) \quad (\text{Figure 3c}), \quad (14b) $$

$$ K_3 = \text{cone}\left(\text{co}\left(\{(1/2, \sqrt{3}/2), (1/2, -\sqrt{3}/2)\}\right)\right) \quad (\text{Figure 3e}). \quad (14c) $$

**Examples with the rotund unit ball of the $ℓ_2$ norm $||·||_2$.** Let us consider the source norm $||·|| = ||·||_2$. We recall that, with this choice of source norm, the unit ball $B$ is rotund, so that we can rely on Corollary 2.5 to characterize Capra-convex sets.

We discuss the case of the three cones {$K_i$}$_{i \in \{1,2,3\}}$ introduced in (14) and of the closed convex hull of their image by the radial projection $\rho$ in (4), under the source norm $||·|| = ||·||_2$. These sets are illustrated in Figure 3.

For all three cones {$K_i$}$_{i \in \{1,2,3\}}$, we have that $K_i \cup \{0\}$ is closed and $0 \notin K_i$. It follows that, according to Corollary 2.5, we only have to check the condition $\overline{\text{co}}(\rho(K_i)) \cap \{0\} = \emptyset$ to assert the Capra-convexity of these cones.

* For the cone $K_1$ in Figure 3a, we have that $\overline{\text{co}}(\rho(K_1)) \cap \{0\} = \{0\}$ in Figure 3b. We conclude from Corollary 2.5 that $K_1$ is not Capra-convex for the choice of source norm $||·|| = ||·||_2$.

* For the cone $K_2$ in Figure 3c, we have that $\overline{\text{co}}(\rho(K_2)) \cap \{0\} = \emptyset$ in Figure 3d. We conclude from Corollary 2.5 that $K_2$ is Capra-convex for the choice of source norm $||·|| = ||·||_2$.

* For the cone $K_3$ in Figure 3e, we have that $\overline{\text{co}}(\rho(K_3)) \cap \{0\} = \emptyset$ in Figure 3f. We conclude from Corollary 2.5 that $K_3$ is Capra-convex for the choice of source norm $||·|| = ||·||_2$.

The example of $K_2$ in Figure 3c reveals that a cone needs not be convex to be Capra-convex. Also, we notice that the Capra-convexity of $K_3$ in Figure 3e can be directly deduced from Corollary 2.6, as $K_3$ is a closed convex pointed cone.

**Examples with the nonrotund unit ball of the $ℓ_\infty$ norm $||·||_\infty$.** Let us now consider the source norm $||·|| = ||·||_\infty$. We recall that, with this choice of source norm, the unit ball $B$ is not rotund, so that we rely mostly on Theorem 2.3 and Corollary 2.6 to characterize Capra-convex sets.

We discuss the case of the three cones {$K_i$}$_{i \in \{1,2,3\}}$ introduced in (14) and of the closed convex hull of their image by the radial projection $\rho$ in (4), under the source norm $||·|| = ||·||_\infty$. These sets are illustrated in Figure 4.

* For the cone $K_1$ in Figure 4a, we have that $K_1 \cap S_\infty^{(0)} = \{(1,0), (-1,1), (-1,-1)\}$, whereas the intersection of $\overline{\text{co}}(\rho(K_1))$ in Figure 4b with $S_\infty^{(0)}$ gives a larger set — which
<figure><img src="image_4.png" /><figcaption>(a) $K_1$ is not Capra-convex for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b) $0 \in \overline{\text{co}}(\varrho(K_1))$ for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(c) $K_2$ is Capra-convex for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(d) $0 \notin \overline{\text{co}}(\varrho(K_2))$ for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(e) $K_3$ is Capra-convex for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(f) $0 \notin \overline{\text{co}}(\varrho(K_3))$ for $\|\cdot\| = \|\cdot\|_2$</figcaption></figure>

<caption>Figure 3: The cones $\{K_i\}_{i \in \{1,2,3\}}$ in (14) (left column) and the closed convex hull of their image by the radial projection $\varrho$ in (4) defined with the source norm $\|\cdot\| = \|\cdot\|_2$ (right column)</caption>
contains for instance the origin 0. We conclude from Theorem 2.3 that $K_1$ is not Capra-convex for the choice of source norm $\|\cdot\| = \|\cdot\|_{\infty}$.

* For the cone $K_2$ in Figure 4c, we have that $K_2 \cap S_{\infty}^{(0)} = \{(-1, 0), (-1, 1), (-1, -1)\}$ whereas the intersection of $\overline{\text{co}}(\rho(K_2)) = \text{co}(\{(-1, 1), (-1, -1)\})$ in Figure 4d with $S_{\infty}^{(0)}$ gives the set $\text{co}(\{(-1, 1), (-1, -1)\})$. We conclude from Theorem 2.3 that $K_2$ is not Capra-convex for the choice of source norm $\|\cdot\| = \|\cdot\|_{\infty}$.

* For the cone $K_3$ in Figure 4e, we have that $K_3 \cap S_{\infty}^{(0)} = \overline{\text{co}}(\rho(K_3)) \cap S_{\infty}^{(0)}$ (see the representation of $\overline{\text{co}}(\rho(K_3))$ in Figure 4f). We conclude from Theorem 2.3 that $K_3$ is Capra-convex for the choice of source norm $\|\cdot\| = \|\cdot\|_{\infty}$.

It is interesting to notice that the cone $K_2$ is not Capra-convex for the source norm $\|\cdot\| = \|\cdot\|_{\infty}$ but is Capra-convex for the source norm $\|\cdot\| = \|\cdot\|_2$. In particular, we observe that the conditions in the right-hand side of (10) are still fulfilled when $\|\cdot\| = \|\cdot\|_{\infty}$, since, in this case, we have that $K_2 \cap \{0\} = \overline{\text{co}}(\rho(K_2)) \cap \{0\} = \emptyset$ (see the representation of $\overline{\text{co}}(\rho(K_2))$ in Figure 4d). This example highlights that the characterization (10) in Corollary 2.5 is not sufficient to identify a Capra-convex set when the unit ball induced by the source norm is not rotund, as in the case of $\|\cdot\| = \|\cdot\|_{\infty}$.

Lastly, as with the source norm $\|\cdot\| = \|\cdot\|_2$, the Capra-convexity of $K_3$ in Figure 4e can be directly deduced from Corollary 2.6, as $K_3$ is a closed convex pointed cone. Indeed, we recall that there is no assumption on the source norm made in Corollary 2.6.

## 3.2 The sublevel sets of Capra-convex functions

As in usual convexity, the Definition 2.1 of Capra-convex sets implies that the sublevel sets of Capra-convex functions are Capra-convex sets. Those sublevel sets appear in cardinality-constrained problems [18] through the Capra-convex function $l_0$ [2]. This is however not the case for the epigraphs of Capra-convex functions. We gather these two results in the following proposition.

We consider the mapping $\theta: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n \times \mathbb{R}$ given by $\theta(x, t) = (\rho(x), t)$, for any $(x, t) \in \mathbb{R}^n \times \mathbb{R}$. Then, introducing the coupling function $\star_{\theta}: (\mathbb{R}^n \times \mathbb{R}) \times (\mathbb{R}^n \times \mathbb{R}) \to \mathbb{R}$ given by $\star_{\theta}((x, t), (y, s)) = \langle \theta(x, t) | (y, s) \rangle$, for any $((x, t), (y, s)) \in (\mathbb{R}^n \times \mathbb{R}) \times (\mathbb{R}^n \times \mathbb{R})$. We say that a set $X \subset \mathbb{R}^n \times \mathbb{R}$ is $\star_{\theta}$-convex if $l_X = l_X^{\star_{\theta}\star_{\theta'}}$.

**Proposition 3.1** Let $\|\cdot\|$ be a source norm, and $\star$ be the corresponding Capra-coupling as in Definition 1.2. Let $f: \mathbb{R}^n \to \mathbb{R}$ be a Capra-convex function, as in Definition 1.3. Then

(i) the sublevel sets $f^{\le t} = \{x \in \mathbb{R}^n \mid f(x) \le t\}$ are Capra-convex sets, for any $t \in \mathbb{R}$;

(ii) the epigraph $\text{epif} = \{(x, t) \in \mathbb{R}^n \times \mathbb{R} \mid f(x) \le t\}$ is a $\star_{\theta}$-convex set.

**Proof.** According to [2, Proposition 2.6], there exists a proper l.s.c. convex function $F: \mathbb{R}^n \to \overline{\mathbb{R}}$ such that $f = F \circ \rho$. Thus, $f(x) \le t \iff F(\rho(x)) \le t$, for any $x \in \mathbb{R}^n$ and $t \in \mathbb{R}$.
<figure><img src="image_10.png" /><figcaption>(a) $K_1$ is not Capra-convex for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(b) $\overline{\text{co}}(\varrho(K_1))$ for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(c) $K_2$ is not Capra-convex for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(d) $\overline{\text{co}}(\varrho(K_2))$ for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(e) $K_3$ is Capra-convex for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(f) $\overline{\text{co}}(\varrho(K_3))$ for $\|\cdot\| = \|\cdot\|_{\infty}$</figcaption></figure>

Figure 4: The cones $\{K_i\}_{i \in \{1,2,3\}}$ in (14) (left column) and the closed convex hull of their image by the radial projection $\varrho$ in (4) defined with the source norm $\|\cdot\| = \|\cdot\|_{\infty}$ (right column)
Thus, elementary calculus rules on indicators functions yield that

$$ \begin{aligned} \iota_{f \le t} &= \iota_{\varrho^{-1}(F \le t)} = \iota_{F \le t} \circ \varrho, \quad \forall t \in \mathbb{R}, \\ \iota_{\text{epi}f} &= \iota_{\theta^{-1}(\text{epi}F)} = \iota_{\text{epi}F} \circ \theta. \end{aligned} $$

Applying the reverse implication in [2, Proposition 2.6] — that a set $X \subset \mathbb{R}^n \times \mathbb{R}$ is $\star_{\theta}$-convex if $\iota_X = G \circ \theta$, where $G: \mathbb{R}^n \to \bar{\mathbb{R}}$ is a proper l.s.c. convex function — we conclude that $\iota_{f \le t}$ is Capra-convex, for any $t \in \mathbb{R}$. This proves (i). Using the same argument, $\iota_{\text{epi}f}$ is a $\star_{\theta}$-convex set, as $\iota_{\text{epi}F}$ is a proper l.s.c. convex function. This proves (ii). $\square$

**Remark 3.2** For a given source norm $\|\cdot\|$ and its associated radial projection $\varrho: \mathbb{R}^n \to S^{(0)}$, there is no hope to rewrite the mapping $\theta: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n \times \mathbb{R}$ defined by $\theta(x, t) = (\varrho(x), t)$ into the radial projection of an other norm. Indeed, the mapping $\theta$ is not 0-homogeneous, while radial projections are.

# 4 Conclusion

We have given a definition for Capra-convex sets, together with results on the characterization of such sets, which are cones that need not be either convex nor closed. Interestingly, we have obtained that the Capra-convexity of a set depends on the geometry of the unit ball induced by the source norm used to define the Capra-coupling. To complete these results, we have provided conditions for the conical hull of a set to be Capra-convex. These conditions allow to characterize sparse optimization problems which could be addressed in the framework of Capra-convexity. In particular, we have showed that the conical hull of a closed spherically-convex set is Capra-convex. Lastly, we have provided several examples to illustrate the specificity of Capra-convex sets.

# A Appendix: proofs

## A.1 Generic lemmas

We show a first characterization of Capra-convex sets.

**Lemma A.1** Let $\|\cdot\|$ be a source norm on $\mathbb{R}^n$, $\varrho$ be the corresponding Capra-coupling in (6), and $\varrho$ be the corresponding radial projection in (4). Let $X \subseteq \mathbb{R}^n$ be a set. The following assertions are equivalent:

(i) $X$ is Capra-convex,

(ii) $\iota_X = \iota_{\overline{\text{co}}(\varrho(X))} \circ \varrho$,

(iii) $X = \varrho^{-1}(\overline{\text{co}}(\varrho(X)))$.
**Proof.** To begin with, we show the following preliminary result: $\iota_X^{\psi} = \iota_{\overline{\text{co}}}(\varrho(X)) \circ \varrho$. The coupling $\psi$ in Definition 1.2 is one-sided linear (see [2, Definition 2.3]) and factorizes with the radial projection $\varrho: \mathbb{R}^n \to S^{(0)}$ in (4). Therefore, we get that

$$ \begin{aligned} \iota_X^{\psi} &= \iota_X^{\psi'} \circ \varrho, && (\text{as } \psi \text{ is a one-sided linear coupling, see [2, Proposition 2.5]}) \\ &= (\sigma_{\varrho(X)})^{\psi'} \circ \varrho, && (\text{from } \iota_X^{\psi} = \sigma_{\varrho(X)}, \text{ see [2, Proposition 2.5]}) \\ &= (\sigma_{\overline{\text{co}}(\varrho(X))})^{\psi'} \circ \varrho, && (\text{see e.g. [1, Proposition 7.13]}) \\ &= (\iota_{\overline{\text{co}}(\varrho(X))})^{\psi''} \circ \varrho, && (\text{see e.g. [1, Example 13.3]}) \\ &= \iota_{\overline{\text{co}}(\varrho(X))} \circ \varrho. && (\text{as } \overline{\text{co}}(\varrho(X)) \text{ is closed and convex}) \end{aligned} $$

Then, from the characterization of Capra-convex functions in Definition 1.3 and Capra-convex sets in Definition 2.1, we obtain that the set $X$ is Capra-convex iff $\iota_X = \iota_X^{\psi} = \iota_{\overline{\text{co}}(\varrho(X))} \circ \varrho$. This proves (i) $\iff$ (ii). The equivalence between (ii) and (iii) is immediate as $\iota_{\overline{\text{co}}(\varrho(X))} \circ \varrho = \iota_{\overline{\text{co}}(\varrho(X))}$. $\square$

Additionally, we recall the following lemma, which is given as an exercise in [15, Exercise 3.48(a)].

**Lemma A.2** Let $X \subseteq \mathbb{R}^n$ be a compact set such that $0 \notin X$. The set $\text{cone}(X) \cup \{0\}$ is closed.

## A.2 Proof of Theorem 2.3

**Proof.** We show the equivalence in (9) as two reverse implications.

( $\implies$ ) Let us assume that $K$ is Capra-convex. We notice that, following Proposition 2.2, the set $K$ is a cone. Then, we have that

$$ \begin{aligned} K &= \varrho^{-1}(\overline{\text{co}}(\varrho(K))), && (\text{by Item (iii) in Lemma A.1}) \\ &= \text{cone}(\overline{\text{co}}(\varrho(K)) \cap S^{(0)}), && (\text{from (5d)}) \end{aligned} $$

from which we deduce that

$$ \varrho(K) = \text{cone}(\overline{\text{co}}(\varrho(K)) \cap S^{(0)}) \cap S^{(0)} = \overline{\text{co}}(\varrho(K)) \cap S^{(0)}. \quad (\text{from (5c), as } K \text{ is a cone}) $$

( $\Leftarrow$ ) Let us assume that $K$ is a cone and $\varrho(K) = \overline{\text{co}}(\varrho(K)) \cap S^{(0)}$. We have that

$$ \begin{aligned} K &= \varrho^{-1}(\varrho(K)) && (\text{by definition (4) of the radial projection } \varrho, \text{ and as } K \text{ is a cone}) \\ &= \varrho^{-1}(\overline{\text{co}}(\varrho(K)) \cap S^{(0)}) && (\text{by assumption}) \\ &= \varrho^{-1}(\overline{\text{co}}(\varrho(K))). && (\text{from (5d)}) \end{aligned} $$

We conclude that that $K$ is Capra-convex by Item (iii) in Lemma A.1. $\square$
## A.3 Proof of Corollary 2.4

**Proof.** Let us suppose that the set $K$ is Capra-convex, and prove the implication ($\implies$) in (10).

* As shown in Proposition 2.2, $K$ is a cone.

* Since $K$ is a cone, we have that $K \cup \{0\} = \text{cone}(K \cap S) \cup \{0\}$. Then, from Theorem 2.3, as $K$ is Capra-convex, we obtain $K \cup \{0\} = \text{cone}(\overline{\text{co}}(K \cap S^{(0)}) \cap S) \cup \{0\}$. Lastly, as $\overline{\text{co}}(K \cap S^{(0)}) \cap S$ is a compact set which does not contain 0, we conclude from Lemma A.2 that $K \cup \{0\}$ is closed.

* The fact that $K \cap \{0\} = \overline{\text{co}}(\varrho(K)) \cap \{0\}$ follows from

$$ 
\begin{align*}
0 \in K &\iff \iota_K(0) = 0, \\
&\iff \iota_{\overline{\text{co}}(\varrho(K))}(0) = 0, \tag{from Lemma A.1, as $\varrho(0) = 0$} \\
&\iff 0 \in \overline{\text{co}}(\varrho(K)).
\end{align*}
$$

This ends the proof of the implication ($\implies$). $\square$

## A.4 Proof of Corollary 2.5 (rotund norm balls)

**Proof.** We now prove that if the unit ball $B$ of the source norm is rotund, then the reverse implication ($\Leftarrow$) in (10) holds.

Let us assume that the set $K$ satisfies the three conditions in the right-hand side of the implication in (10), that is,

$$ 
\left\{ 
\begin{array}{l}
K \text{ is a cone,} \\
K \cup \{0\} \text{ is closed,} \\
K \cap \{0\} = \overline{\text{co}}(\varrho(K)) \cap \{0\}.
\end{array}
\right. 
$$

We show that, under these assumptions, $K \cap S^{(0)} = \overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)}$, hence that (9) is satisfied, from (5c).

The inclusion $K \cap S^{(0)} \subseteq \overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)}$ is straightforward. We thus concentrate on the reverse inclusion. Let us take $x \in \overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)}$. We consider two cases.

* Let us assume that $x = 0$. We deduce that $0 \in \overline{\text{co}}(K \cap S^{(0)}) = \overline{\text{co}}(\varrho(K))$, since $K$ is a cone (see (5c)). Then, as $K \cap \{0\} = \overline{\text{co}}(\varrho(K)) \cap \{0\}$, we conclude that $0 \in K$, and thus that $x = 0 \in K \cap S^{(0)}$.

* We now turn to the case $x \neq 0$. We observe that

$$ K \cap S^{(0)} = \begin{cases} (K \cup \{0\}) \cap S^{(0)}, & \text{if } 0 \in K, \\ (K \cup \{0\}) \cap S, & \text{if } 0 \notin K. \end{cases} $$

Since $K \cup \{0\}$ is closed, we deduce that $K \cap S^{(0)}$ is closed, and thus compact (since included in the unit ball $B$). It follows that the convex hull of $K \cap S^{(0)}$ is compact (see e.g. [15, Corollary 2.30]), and thus that $\overline{\text{co}}(K \cap S^{(0)}) = \text{co}(K \cap S^{(0)})$.

We deduce that $x \in \text{co}(K \cap S^{(0)})$, and therefore, from Carathéodory's Theorem (see e.g. [15, Theorem 2.29]), that there exists $\{\alpha_i\}_{i \in \llbracket 1, n+1 \rrbracket} \in [0, 1]^{n+1}$, $\{x_i\}_{i \in \llbracket 1, n+1 \rrbracket} \in (K \cap S^{(0)})^{n+1}$ such that

$$ x = \sum_{i=1}^{n+1} \alpha_i x_i, \quad \text{and} \quad \sum_{i=1}^{n+1} \alpha_i = 1. $$
Now, we observe that if for some $j \in [1, n+1]$ we have $x_j = 0$ and $\alpha_j > 0$, as $x \in S$, it implies that

$$1 = \|x\| \le \sum_{i=1}^{n+1} \alpha_i \|x_i\| \le \sum_{i \ne j, i=1}^{n+1} \alpha_i < 1,$$

which leads to a contradiction. We deduce that $\{x_i\}_{i \in [1, n+1]} \in (K \cap S)^{n+1}$, and thus that $x \in \text{co}(K \cap S)$. Finally, from [5, Corollary 16], since the unit ball induced by $\|\cdot\|$ is rotund, we have that $\text{co}(K \cap S) \cap S = K \cap S$, so that $x \in K \cap S$.

We conclude that, in both cases, $x \in K \cap S^{(0)}$, which proves that $K \cap S^{(0)} \supset \overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)}$, and therefore that $K \cap S^{(0)} = \overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)}$.

Therefore, (9) holds and we deduce from Theorem 2.3 that the cone $K$ is a Capra-convex set.

□

## A.5 Proof of Corollary 2.6 (closed convex cones)

**Proof.** Let $K \subseteq \mathbb{R}^n$ be a closed convex cone.

First, we prove (i) by showing that $\overline{\text{co}}(K \cap S^{(0)}) = K \cap B$. As $K \cap S^{(0)} \subset K \cap B$, we have that $\overline{\text{co}}(K \cap S^{(0)}) \subseteq \overline{\text{co}}(K \cap B)$. It follows that $\overline{\text{co}}(K \cap S^{(0)}) \subseteq K \cap B$ from the fact that $\overline{\text{co}}(K \cap B) = K \cap B$, since $K \cap B$ is closed convex.

To prove the reverse inclusion, let us consider $x \in K \cap B$. By definition of the radial projection $\varrho$ in (4), we have that $x = \|x\|\varrho(x) + (1 - \|x\|)0$ with $\{\varrho(x), 0\} \subseteq K \cap S^{(0)}$ as $x \in K$ and $0 \in K$, and $\|x\| \le 1$, as $x \in B$. We deduce that $x \in \text{co}(K \cap S^{(0)})$, which proves the reverse inclusion $\overline{\text{co}}(K \cap S^{(0)}) \supset K \cap B$.

Now, we observe that $\overline{\text{co}}(K \cap S^{(0)}) \cap S^{(0)} = K \cap B \cap S^{(0)} = K \cap S^{(0)}$ which gives that $\varrho(K) = \overline{\text{co}}(\varrho(K)) \cap S^{(0)}$, from (5c), as $K$ is a cone. We obtain that $K$ is Capra-convex, from Theorem 2.3.

Second, we assume that, moreover, the cone $K$ is pointed, and we prove (ii). To proceed, we introduce the notation $K' = K \setminus \{0\}$ and we show that $K' \cap S^{(0)} = \overline{\text{co}}(K' \cap S^{(0)}) \cap S^{(0)}$. The inclusion $K' \cap S^{(0)} \subseteq \overline{\text{co}}(K' \cap S^{(0)}) \cap S^{(0)}$ is straightforward. We thus concentrate on the reverse inclusion.

Let us take $x \in \overline{\text{co}}(K' \cap S^{(0)}) \cap S^{(0)}$. As $K' \cap S^{(0)} = K \cap S$ is a compact set, so is its convex hull (see e.g. [15, Corollary 2.30]), and thus $\overline{\text{co}}(K' \cap S^{(0)}) = \text{co}(K \cap S)$. We therefore have $x \in \text{co}(K \cap S) \cap S^{(0)} \subset \text{co}(K) \cap S^{(0)} = K \cap S^{(0)}$, as $K$ is convex.

We now assume that $x = 0$, and show that it leads to a contradiction. As $x \in \text{co}(K \cap S)$, we obtain, from Carathéodory's Theorem (see e.g. [15, Theorem 2.29]), that there exists $\{\alpha_i\}_{i \in [1, n+1]} \in [0, 1]^{n+1}$, $\{x_i\}_{i \in [1, n+1]} \in (K \cap S)^{n+1}$ such that

$$0 = x = \sum_{i=1}^{n+1} \alpha_i x_i, \quad \text{and} \quad \sum_{i=1}^{n+1} \alpha_i = 1.$$

Thus, there necessarily exist $j \in [1, n+1]$ such that $\alpha_j \ne 0$. By construction, $x_j \in K$ and

$$-x_j = \sum_{i \in [1, n+1], i \ne j} \frac{\alpha_i}{\alpha_j} x_i.$$
As $K$ is a convex cone, we deduce from the above expression that $-x_j \in K$ (see e.g. [1, Proposition 6.3(i)]). Then, since $K$ is pointed, necessarily $x_j = 0$, which contradicts the fact that $x_j \in S$. We conclude that $x \neq 0$, and therefore that $x \in K \cap S = K' \cap S^{(0)}$.

This proves the reverse inclusion $K' \cap S^{(0)} \supset \overline{\text{co}}(K' \cap S^{(0)}) \cap S^{(0)}$. Finally, we have obtained that $K' \cap S^{(0)} = \overline{\text{co}}(K' \cap S^{(0)}) \cap S^{(0)}$ and the conclusion follows (5c), as $K'$ is a cone. This concludes the proof. $\square$

## A.6 Proof of Proposition 2.7

**Proof.** Let $X \subseteq \mathbb{R}^n$ be a compact set such that $0 \notin \text{co}(X)$, hence $0 \notin X$. Following Lemma A.2, the set $\text{cone}(X) \cup \{0\}$ is closed.

* Let us assume (i).

We start by proving that $0 \notin \overline{\text{co}}(\varrho(\text{cone}(X)))$ by contradiction. Assume that $0 \in \overline{\text{co}}(\varrho(\text{cone}(X)))$. As $0 \notin X$ by assumption, and by definition (2a) of the conical hull, we have that $0 \notin \text{cone}(X)$, so that, using (5c), we deduce that $0 \in \overline{\text{co}}(\text{cone}(X) \cap S)$ and thus $0 \in \text{co}(\text{cone}(X) \cap S)$ since $\text{cone}(X) \cap S$ is compact (see e.g. [15, Corollary 2.30]). We obtain from Carathéodory's Theorem (see e.g. [15, Theorem 2.29]) that there exists $\{\alpha_i\}_{i \in [1, n+1]} \in [0, 1]^{n+1}$, $\{x_i\}_{i \in [1, n+1]} \in (\text{cone}(X) \cap S)^{n+1}$ such that

$$0 = \sum_{i=1}^{n+1} \alpha_i x_i, \quad \text{and} \quad \sum_{i=1}^{n+1} \alpha_i = 1.$$

By definition of the conical hull in (2a), there exists $\{x'_i\}_{i \in [1, n+1]} \in X^{n+1}$ such that $x_i = \varrho(x'_i)$ for $i \in [1, n+1]$. Thus, introducing $\theta = \sum_{i=1}^{n+1} \alpha_i / \|x'_i\| > 0$, we have

$$0 = \sum_{i=1}^{n+1} \frac{\alpha_i}{\theta \|x'_i\|} x'_i, \quad \text{and} \quad \sum_{i=1}^{n+1} \frac{\alpha_i}{\theta \|x'_i\|} = 1,$$

which implies that $0 \in \text{co}(X)$ and leads to a contradiction.

We conclude that $0 \notin \overline{\text{co}}(\varrho(\text{cone}(X)))$ and, therefore, that $X$ is Capra-convex, from the reciprocal implication in (10) in Corollary 2.5 in the case of a rotund unit norm ball.

* Now let us assume (ii).

Consider the cone $K = \text{cone}(X) \cup \{0\} = \text{pos}(X)$ which is convex as $X$ is convex. Moreover, we have that $0 \notin X$, as $X$ is convex and $0 \notin \text{co}(X)$ which, combined with Lemma A.2 and the fact that $X$ is compact, gives that $K$ is closed. Thus $K$ is a closed convex cone. If we prove now that $K$ is a pointed cone, then we will obtain from Corollary 2.6 that $\text{cone}(X) = K \setminus \{0\}$ is a Capra-convex set.

To conclude the proof, it remains to show that $K$ is a pointed cone. We prove the result by contradiction. Assume that there exists $x \in \text{cone}(X) \cap (-\text{cone}(X))$. Then, there exists $\lambda > 0$ and $\beta > 0$ and $x_1, x_2$ in $X$ such that $x = \lambda x_1$ and $x = -\mu x_2$. We therefore obtain that $0 = (\lambda x_1 + \mu x_2)/(\lambda + \mu)$ and therefore that $0 \in \text{co}(X)$ which leads to a contradiction. We conclude that $K \cap (-K) = \{0\}$ and, therefore, that $K$ is a pointed cone.

This concludes the proof. $\square$
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