Hi,

This is a problem that has being bothering me the last few days.

Assume a convex function $f(x): {\mathbb R}^n \rightarrow {\mathbb R}$ with a unique minimizer $x^{\star}$. Now consider the problem of minimizing $(1-\gamma) f(x) + \gamma f(-x)$, for some small $0< \gamma < 1 $. Say that the minimizer of this (convex) problem is $x^{\star \star}$.

Do you see any apparent way of bounding the Euclidean distance between $x^{\star}$ and $x^{\star \star}$?

What are your thoughts on the matter?