Strictly convex function in two variables, do the points where the function is partially optimized in one variable trace out a straight line?

Question

Suppose $f(x,y):\mathbb{R}^2 \rightarrow \mathbb{R}$ is strictly convex in $(x,y)$. It is known that the function:

$g(x)=\inf_{y}f(x,y)=\min_{y}f(x,y)$

is convex in $x$, provided $g(x) > -\infty$ for all $x$. However, I am wondering if it is also the case that the set:

$H=\{(x,y):y=argmin_{y}f(x,y)\}$

is necessarily a line in $\mathbb{R}^2$. I think it's impossible for $H$ to trace out a curve while still preserving convexity, but it seems difficult to prove.

Answer 1

No, consider $$f(x,y) = (x-y)^2 + e^{x+y}$$ Three approximate points on $H$ are $(-1,-1.06)$, $(0,-0.37)$ and $(1,-0.15)$, which are not on a line.