Theorem: Fan-Glicksberg-Hoffman Concave Alternative

Let $X$ be a vector space over $\R$.
Let $C \subseteq X$ be a convex set.
Let $f_1, \ldots, f_m : C \to \R$ be concave functions.
Then exactly one of the following is true:
:$(1): \quad$ there exists $x \in C$ such that $\map {f_i} x > 0$ for each $1 \le i \le n$
:$(2): \quad$ there exists $\lambda_1, \ldots, \lambda_n \in \R_{\ge 0}$, not all zero, such that:
::::$\ds \sum_{i \mathop = 1}^n \lambda_i \map {f_i} x \le 0$ for each $x \in C$.

Proof:
We show that $(1)$ and $(2)$ cannot hold simultaneously.
Suppose that $(1)$ holds.
Then:
:there exists $x \in C$ such that $\map {f_i} x > 0$ for each $1 \le i \le n$.
Let $\lambda_1, \ldots, \lambda_n \in \R_{\ge 0}$ be such that $\lambda_j \ne 0$. 
Then we have:
:$\ds \sum_{i \mathop = 1}^n \lambda_i \map {f_i} x \ge \lambda_j \map {f_j} x > 0$
Hence $(2)$ does not hold.
Conversely suppose that $(2)$ holds.
Then there exists $\lambda_1, \ldots, \lambda_n \in \R_{\ge 0}$ not all zero such that:
:$\ds \sum_{i \mathop = 1}^n \lambda_i \map {f_i} x \le 0$ for each $x \in C$.
Hence there exists $1 \le j \le n$ such that $\map {f_j} x < 0$.
Hence $(1)$ does not hold. 
We now show that either $(1)$ or $(2)$ holds.
Suppose that $(1)$ is not true.
Let:
:$A = \set {y \in \R^m : \text {there exists } x \in C \text { such that } y_i \le \map {f_i} x \text { for each } i}$
For each $x \in C$ we have:
:$\tuple {\map {f_1} x, \ldots, \map {f_m} x} \in A$ for each $x \in C$.
Hence $A \ne \O$.
We show that $A$ is convex.
Let $y, z \in A$ and $\alpha \in \closedint 0 1$.
Then there exists $x_1, x_2 \in C$ such that:
:$y_i \le \map {f_i} {x_1}$ for each $1 \le i \le m$
and:
:$z_i \le \map {f_i} {x_2}$ for each $1 \le i \le m$. 
We have:
{{begin-eqn}}
{{eqn	| l = \alpha y_i + \paren {1 - \alpha} z_i
		| o = \le
		| r = \alpha \map {f_i} {x_1} + \paren {1 - \alpha} \map {f_i} {x_2}
}}
{{eqn	| o = \le
		| r = \map {f_i} {\alpha x_1 + \paren {1 - \alpha} x_2}
		| c = {{Defof|Concave Real Function}}
}}
{{end-eqn}}
Since $C$ is convex, we have:
:$\alpha x_1 + \paren {1 - \alpha} x_2 \in C$
Hence:
:$\alpha y + \paren {1 - \alpha} z \in A$
Hence $A$ is a convex set in $\R^m$.
From Set of Vectors in Euclidean Space with all Components Non-Negative is Convex:
:$\R^m_+ = \set {y \in \R^m : y_i \ge 0 \text { for each } i}$ is convex.
From Interior of Set of Vectors in Euclidean Space with all Components Non-Negative, we have:
:$\Int {\R^m_+} = \set {y \in \R^m : y_i > 0 \text { for each } i}$
Since $(1)$ is false, for all $x \in C$ there exists $i$ such that $\map {f_i} x \le 0$. 
Hence:
:$\tuple {\map {f_1} x, \ldots, \map {f_m} x} \not \in \Int {\R^m_+}$ for each $x \in C$.
In particular, if $y \in \R^m$ satisfies $y_i \le \map {f_i} x$ for all $1 \le i \le m$, we have:
:$\tuple {y_1, \ldots, y_m} \not \in \Int {\R^m_+}$
That is:
:$A \cap \Int {\R^m_+} = \O$
By the Interior Separating Hyperplane Theorem, there exists a bounded linear functional $\phi : \R^m \to \R$ such that:
:$\map \phi y \ge \alpha \ge \map \phi u$ for each $y \in \R^m_+$ and $u \in C$
where:
:there exists $x_0 \in C$ such that $\map \phi {x_0} \ne \alpha$. 
Hence $\phi \ne {\mathbf 0}_{\paren {\R^m}^\ast}$. 
From the Riesz Representation Theorem (Hilbert Spaces), there exists $\lambda = \tuple {\lambda_1, \ldots, \lambda_m} \in \R^m$ such that:
:$\lambda \cdot y \ge \lambda \cdot u$
Since $\phi \ne {\mathbf 0}_{\paren {\R^m}^\ast}$, we have $\lambda_i \ne 0$ for some $i$.
Taking:
:$u = \tuple {\map {f_1} x, \ldots, \map {f_m} x}$ for $x \in C$.
We then have:
:$\ds \sum_{i \mathop = 1}^m \lambda_i y_i \ge \sum_{i \mathop = 1}^m \lambda_i \map {f_i} x$ 
for all $x \in C$ and $\tuple {y_1, \ldots, y_m} \in \R^m_+$.
Taking $y = \tuple {0, 0, \ldots, 0}$, we obtain that:
:$\ds \sum_{i \mathop = 1}^m \lambda_i \map {f_i} x \le 0$ for all $x \in C$.
Now {{AimForCont}} that $\lambda_i < 0$ for some $i$.
Take $y = t e_i$ for some $t > 0$.
Then we have:
:$\ds t \lambda_i \ge \sum_{i \mathop = 1}^m \lambda_i \map {f_i} x$ for each $x \in C$.
Taking $t \to \infty$, we obtain:
:$\ds \sum_{i \mathop = 1}^m \lambda_i \map {f_i} x = -\infty$ for each $x \in C$
a contradiction to each $f_i$ taking real values.
Hence $\lambda_i \ge 0$ for each $i$. 
Hence we have shown that:
:$\ds \sum_{i \mathop = 1}^m \lambda_i \map {f_i} x \le 0$ for all $x \in C$
where $\lambda_i \ge 0$ for each $i$.
Further, we have established that $\lambda \ne {\mathbf 0}_{\R^m}$.
This is situation $(2)$. 
Hence we have shown that $(1)$ and $(2)$ cannot hold simultaneously, and further that if $(1)$ does not hold then $(2)$ holds. 
Hence precisely one of $(1)$ or $(2)$ holds.
{{qed}}
{{Namedfor|Ky Fan|name2 = Irving Glicksberg|name3 = Alan Jerome Hoffman|cat = Fan|cat2 = Glicksberg|cat3 = Hoffman}}