I am wondering whether this is true or false, since I can't seem to find any counterexamples.
Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is convex and has a minimizer $\mathbf{x}^{\star}$. Let $\mathbf{x}\in\mathbb{R}^d$ and consider the closed ball of radius $r$ around $\mathbf{x}$, denoted $B_r(\mathbf{x})$. Then $f$ is minimized at the point on the boundary of $B_r(\mathbf{x})$ closest to $\mathbf{x}^{\star}$. (Of course, we assume $\mathbf{x}^{\star}$ is not in the ball itself.)
It seems intuitive, but I can't seem to use convexity nicely. Nor have I found any counterexamples yet. Any suggestions?