The Exercise

A real function of two variables can be concave in two ways. It can be concave in each variable separately, as a function of $x$ for each fixed $y$ and of $y$ for each fixed $x$ , or concave in the pair $(x,y)$ jointly. The exercise is to show that joint concavity implies concavity in a single variable with the other held fixed, and that the converse fails, with the product $xy$ of two real numbers as the example.

Concave Functions

Let $\Bbbk$ be an ordered field, such as $\mathbb{R}$ , and $C$ a convex set in a vector space over $\Bbbk$ . A function $g$ on $C$ , with values in $\mathbb{R}$ or in any ordered vector space over $\Bbbk$ , is concave on $C$ if $g\bigl(\lambda x_1 + (1-\lambda) x_2\bigr) \ge \lambda\, g(x_1) + (1-\lambda)\, g(x_2)$ (1) for all points $x_1, x_2$ of $C$ and all $\lambda \in \Bbbk$ with $0 \le \lambda \le 1$ .

Vector spaces $E$ and $F$ over $\Bbbk$ carry a set $s$ in $E$ and a set $t$ in $F$ . A function $f$ on $s \times t$ is jointly concave if it is concave on $s \times t$ , the points being the pairs $(x,y)$ with $x \in s$ and $y \in t$ . The function $f$ is concave in its first variable if for every $y \in t$ the slice $x \mapsto f(x,y)$ is concave on $s$ , and concave in its second variable if for every $x \in s$ the slice $y \mapsto f(x,y)$ is concave on $t$ .

The product function $m$ on the plane $\mathbb{R}^2$ is the real function $m(x,y) = xy.$ (2)^✦

The Claim

Theorem 1. Let $f$ be a jointly concave function on $s \times t$ . Then $f$ is concave in its first variable: for every $y \in t$ , $f\bigl(\lambda x_1 + (1-\lambda) x_2,\, y\bigr) \ge \lambda\, f(x_1, y) + (1-\lambda)\, f(x_2, y)$ (3)^✦ for all $x_1, x_2 \in s$ and all $0 \le \lambda \le 1$ .

Theorem 2. The product $m$ is concave in each variable: for all real $x$ , $y$ , $x_1$ , $x_2$ , $y_1$ , $y_2$ and all $0 \le \lambda \le 1$ , $m\bigl(\lambda x_1 + (1-\lambda) x_2,\, y\bigr) \ge \lambda\, m(x_1, y) + (1-\lambda)\, m(x_2, y)$ (4)^✦ and $m\bigl(x,\, \lambda y_1 + (1-\lambda) y_2\bigr) \ge \lambda\, m(x, y_1) + (1-\lambda)\, m(x, y_2).$ (5)^✦ It is not jointly concave: there are points $(x_1,y_1)$ and $(x_2,y_2)$ of the plane and a weight $0 \le \lambda \le 1$ with $m\bigl(\lambda x_1 + (1-\lambda) x_2,\, \lambda y_1 + (1-\lambda) y_2\bigr) < \lambda\, m(x_1, y_1) + (1-\lambda)\, m(x_2, y_2).$ (6)^✦

What Is Checked

About these the library proves (3) for every such function and every point of $t$ , (4) and (5) for all real arguments of $m$ , and the existence of the points and the weight in (6).