Joint maximizer of a strongly concave function

Question

I have a question that is arising in my research.

Suppose that $f : \mathbb{R}^ 2 \to \mathbb{R}$ is a strongly concave function, satisfying:

1. For every $x$, the function $y \to f(x, y)$ is maximized globally at a unique point $$y_0 \in \mathbb{R}$$ depending on $x$.

2. For every $y$, the function $x \to f(x, y)$ is maximized globally at a unique point $$x_0 \in \mathbb{R}$$ depending on $y$.

Can anyone tell me if there is a sufficient condition for the entire function $f$ to be maximized exactly at a unique point $(x_1,y_1)\in \mathbb{R}^2$? In my case, the function is differentiable sufficiently many times.

Answer 1

$\newcommand\R{\mathbb R}$It suffices that $f$ just be strongly concave (your conditions 1 and 2 then hold automatically).

Indeed, then
$$f(y)\le f(x)+h\cdot(y-x)-m|y-x|^2/2$$ for some $x\in\R^2$, some $h\in\R^2$, some real $m>0$, and all $y\in\R^2$, where $\cdot$ denotes the dot product and $|\cdot|$ denotes the Euclidean norm. So, $f(y)<f(x)$ if $|y-x|>2|h|/m$. So, the function $f$, being concave and hence continuous, attains its maximum at some point at distance $\le2|h|/m$ from $x$. Moreover, the maximizer of $f$ is unique, since $f$ is strongly concave and hence strictly concave. $\quad\Box$