# Between Quasi-convex and Convex Set-valued Mappings

Joël Benoist, Nicolae Popovici

► To cite this version:

Joël Benoist, Nicolae Popovici. Between Quasi-convex and Convex Set-valued Mappings. Applied Mathematics Letters, 2004, 17, pp.245-247. 10.1016/S0893-9659(04)90058-4. hal-00068405

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# Between Quasi-Convex and Convex Set-Valued Mappings

J. BENOIST

LACO, UPRESSA 6090, Department of Mathematics

University of Limoges, 87060 Limoges, France

benoist@unilim.fr

N. POPOVICI

Faculty of Mathematics and Computer Science

Babeş-Bolyai University of Cluj, 3400 Cluj-Napoca, Romania

popovici@math.ubbcluj.ro

**Abstract**—The aim of this paper is to give sufficient conditions for a quasi-convex set-valued mapping to be convex. In particular, we recover several known characterizations of convex real-valued functions, given in terms of quasiconvexity and Jensen-type convexity by Nikodem [1], Behringer [2], and Yang, Teo and Yang [3].

**Keywords**—Generalized convexity, Set-valued mappings.

## 1. INTRODUCTION

Throughout the paper, we denote by $X$ a linear space and by $Y$ a topological linear space, partially ordered by a closed convex cone $K$ having a nonempty interior in $Y$. Let $F: D \to 2^Y$ be a set-valued mapping, defined on a nonempty convex subset $D$ of $X$.

Recall that $F$ is said to be $K$-convex if the inclusion

$$tF(x) + (1-t)F(x') \subset F(tx + (1-t)x') + K \quad (1)$$

holds for all $x, x' \in D$ and for every $t \in [0, 1]$.

By analogy to vector-valued functions, we say that $F$ is $K$-quasiconvex if for each $y \in Y$ the level set $L_F(y) := \{x \in D : y \in F(x) + K\}$ is convex. Since $K$ is a convex cone, it can be easily seen that $F$ is $K$-quasiconvex whenever it is $K$-convex.

In order to get sufficient conditions for a $K$-quasi-convex mapping to be $K$-convex, we shall consider the following concept of generalized convexity: $F$ will be called weakly $K$-convex with respect to a nonempty set $T \subset ]0, 1[$ if for all $x, x' \in D$ there exists some $t \in T$ for which (1) holds.

Note that this concept extends several notions of generalized convexity, which were intensively studied in the literature in the particular case of real-valued functions. Indeed, if $T \subset ]0, 1[$ is a
singleton, we recover the Jensen-type convexity (see, e.g., [4] and references therein), which is
nowadays also known as nearly convexity (see, e.g., [5]). On the other hand, for $T = ]0, 1[$ we
recover the notion of weakly convexity, introduced by Aleman in [6]. As an intermediate case, if
$T = [\delta, 1 - \delta]$ with $\delta \in ]0, 1/2[$, we recover the notion of uniform convexlikeness, which has been
introduced by Hartwig in [7].

Our aim here is to study the set-valued mappings, but our main result also focuses on vector-
valued functions. Actually, if $f: D \to Y$ is a function defined on a nonempty convex subset $D$
of $X$, then $f$ will be called $K$-convex (respectively, $K$-quasiconvex, or weakly $K$-convex with
respect to a nonempty set $T \subset ]0, 1[$) if and only if the set-valued mapping $F: D \to 2^Y$, defined
by $F(x) = \{f(x)\}$ for all $x \in D$, is $K$-convex (respectively, $K$-quasiconvex, or weakly $K$-convex
with respect to $T$).

## 2. MAIN RESULT

THEOREM 2.1. Let $F: D \to 2^Y$ be a set-valued mapping defined on a nonempty convex subset $D$ of $X$. If $F$ has $K$-closed values (i.e., $F(x)+K$ is a closed set for every $x \in D$), then the following assertions are equivalent:

(i) $F$ is $K$-convex;

(ii) $F$ is both $K$-quasiconvex and weakly $K$-convex with respect to a nonempty compact set $T \subset ]0, 1[$.

PROOF. Obviously (i) implies (ii), the conclusion being true for any nonempty set $T \subset ]0, 1[$.

Conversely, suppose that (ii) holds and let $T$ be a nonempty compact subset of $]0, 1[$ for which $F$ is weakly $K$-convex. Let us denote, for all $x, x' \in D$,

$$T_{x,x'} := \{t \in [0, 1] : (1) \text{ holds.}\}.$$

In order to prove (i), we just have to show that $T_{x,x'} = [0,1]$ for all $x,x' \in D$. To this end,
consider two arbitrary points $x_0, x_1 \in D$ and let us first prove that $T_{x_0,x_1}$ is dense in $[0,1]$.
Suppose on the contrary that this is not the case. Then there exist some $a, b \in [0,1]$, $a < b$, such
that

$$[a, b] \cap T_{x_0, x_1} = \emptyset. \qquad (2)$$

Since $\{0,1\} \subset T_{x_0,x_1}$, we can define the real numbers

$$\alpha := \sup[0, a] \cap T_{x_0, x_1} \quad \text{and} \quad \beta := \inf[b, 1] \cap T_{x_0, x_1}. \qquad (3)$$

Obviously, $\alpha \le a < b \le \beta$ and, by (2) and (3), we have

$$]\alpha, \beta[ \cap T_{x_0, x_1} = \emptyset. \qquad (4)$$

Let us denote, for all $t \in [0,1]$,

$$x_t := tx_0 + (1-t)x_1 \quad \text{and} \quad Y_t := tF(x_0) + (1-t)F(x_1).$$

By (3) and taking into account that $T$ is compact, we can find some numbers $u \in [0, \alpha] \cap T_{x_0, x_1}$ and $v \in [\beta, 1] \cap T_{x_0, x_1}$ such that $tu + (1-t)v \in ]\alpha, \beta[$ for all $t \in T$. On the other hand, since $F$ is weakly $K$-convex with respect to $T$, we can choose a number $\tau \in T_{x_u, x_v} \cap T$. Hence,

$$\gamma := \tau u + (1 - \tau)v \in ]\alpha, \beta[. \qquad (5)$$

Since $1 - \gamma = \tau(1 - u) + (1 - \tau)(1 - v)$, by definition of $Y_\gamma$ we can deduce that

$$\begin{align}
Y_{\gamma} &= [\tau u + (1 - \tau)v]F(x_0) + [\tau(1 - u) + (1 - \tau)(1 - v)]F(x_1) \nonumber \\
&\subset \tau[uF(x_0) + (1 - u)F(x_1)] + (1 - \tau)[vF(x_0) + (1 - v)F(x_1)] \tag{6} \\
&= \tau Y_u + (1 - \tau)Y_v. \nonumber
\end{align}$$
On the other hand, since $u, v \in T_{x_0, x_1}$, we have $Y_u \subset F(x_u) + K$ and $Y_v \subset F(x_v) + K$, and hence,

$$ \tau Y_u + (1 - \tau)Y_v \subset \tau F(x_u) + (1 - \tau)F(x_v) + K. \quad (7) $$

Recalling that $\tau \in T_{x_u, x_v}$, i.e., $\tau F(x_u) + (1 - \tau)F(x_v) \subset F(\tau x_u + (1 - \tau)x_v) + K$, by (6) and (7) we infer that $Y_\gamma \subset \tau Y_u + (1 - \tau)Y_v \subset F(\tau x_u + (1 - \tau)x_v) + K = F(x_\gamma) + K$, which means that $\gamma \in T_{x_0, x_1}$. By (5) it follows that $\alpha, \beta[\cap T_{x_0, x_1} \neq \emptyset$, contradicting (4).

So, we have proved that $T_{x_0, x_1}$ is dense in $[0, 1]$. Now, let us show that $T_{x_0, x_1} = [0, 1]$. Obviously, $\{0, 1\} \subset T_{x_0, x_1} \subset [0, 1]$. Consider an arbitrary $t \in [0, 1]$. We just need to prove that $t \in T_{x_0, x_1}$, i.e., $Y_t \subset F(x_t) + K$. Let $y \in Y_t$. By definition of $Y_t$ we have $y = tz_0 + (1 - t)z_1$ for some $z_0 \in F(x_0)$ and $z_1 \in F(x_1)$. Consider a point $e \in \text{int } K$. By density of $T_{x_0, x_1}$ in $[0, 1]$, we infer the existence of two sequences: $(t_n^-)_{n \in \mathbb{N}}$ in $T_{x_0, x_1} \cap [0, t]$ and $(t_n^+)_{n \in \mathbb{N}}$ in $T_{x_0, x_1} \cap [t, 1]$, such that

$$ \{y_n^-, y_n^+\} \subset y + \frac{1}{n}e - \text{int } K, \quad \text{for all } n \ge 1, $$

where $y_n^- = t_n^-z_0 + (1 - t_n^-)z_1$ and $y_n^+ = t_n^+z_0 + (1 - t_n^+)z_1$. Then, we have

$$ y + \frac{1}{n}e \in t_n^- F(x_0) + (1 - t_n^-)F(x_1) + \text{int } K \subset F(x_{t_n^-}) + K + \text{int } K \subset F(x_{t_n^+}) + K, $$

$$ y + \frac{1}{n}e \in t_n^+ F(x_0) + (1 - t_n^+)F(x_1) + \text{int } K \subset F(x_{t_n^+}) + K + \text{int } K \subset F(x_{t_n^-}) + K, $$

implying that $\{x_{t_n^-}, x_{t_n^+}\} \subset L_F(y + (1/n)e)$, for all $n \ge 1$. Recalling that $F$ is $K$-quasiconvex and taking into account that $x_t \in [x_{t_n^-}, x_{t_n^+}]$ for each $n \in \mathbb{N}$, we can deduce that

$$ x_t \in L_F \left( y + \frac{1}{n}e \right), \quad \text{i.e.,} \quad y + \frac{1}{n}e \in F(x_t) + K, \quad \text{for all } n \ge 1. $$

Finally, by letting $n \to \infty$, we infer that $y \in \overline{F(x_t) + K} = F(x_t) + K$. ■

**COROLLARY 2.2.** Let $f: D \to Y$ be a function defined on a nonempty convex subset $D$ of $X$. Then $f$ is $K$-convex if and only if it is both $K$-quasiconvex and weakly $K$-convex with respect to a nonempty compact set $T \subset]0, 1[$.

**PROOF.** It follows by Theorem 2.1, where $F: D \to 2^Y$ is defined by $F(x) = \{f(x)\}$ for all $x \in D$. In this case $F(x) + K$ is closed for every $x \in D$, since the cone $K$ is closed. ■

**REMARK 2.3.** The compactness of $T$ is essential. Indeed, consider $X = Y = \mathbb{R}$ and $K = \mathbb{R}_+$, and let $f: D = [0, 1] \to \mathbb{R}$ be defined by $f(x) = 1$ if $x \in [0, 1/2]$, and $f(x) = 0$ if $x \in ]1/2, 1]$. Then $f$ is both quasiconvex and weakly convex with respect to $T =]0, 1[$, but $f$ is not convex.

**REMARK 2.4.** Corollary 2.2 generalizes some known characterization theorems given for real-valued convex functions, such as

(a) Proposition 3 in [1], where $X = \mathbb{R}^n$, $Y = \mathbb{R}$, $K = \mathbb{R}_+$, $D \subset \mathbb{R}^n$ is a nonempty convex open set, and $T = \{1/2\}$;

(b) Theorem 2 in [2], where $X$ is a linear space, $Y = \mathbb{R}$, $K = \mathbb{R}_+$, $D \subset X$ is a nonempty convex set, and $T = \{1/2\}$;

(c) Theorem 3 in [3], where $X = \mathbb{R}^n$, $Y = \mathbb{R}$, $K = \mathbb{R}_+$, $D \subset \mathbb{R}^n$ is a nonempty convex set, and $T = \{\alpha\}$ with $\alpha \in ]0, 1[$.

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2. F.A. Behringer, Convexity is equivalent to midpoint convexity combined with strict quasiconvexity, *Optimization* **24** (3/4), 219-228, (1992).

3. X.M. Yang, K.L. Teo and X.Q. Yang, A characterization of convex function, *Appl. Math. Lett.* **13** (1), 27-30, (2000).

4. A.W. Roberts and D.E. Varberg, *Convex Functions*, Academic Press, New York, (1973).

5. W.W. Breckner and G. Kassay, A systematization of convexity concepts for sets and functions, *J. Convex Anal.* **4** (1), 109-127, (1997).

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