Let $f_1,...,f_m:\mathbb{R}^n\to\mathbb{R}$ be $C^2$ functions. Suppose there exists $\mu>0$ such that $\nabla^2f_i(x)\succeq\mu I$ for all $x\in\mathbb{R}^n$ and for all $i=1,...,m$. That is, $f_1,...,f_m$ are $\mu$-strongly convex. Then each $f_i$ has a unique minimizer $x_i$. I am interested in the location of the minimizer $z$ of $f_1+\cdots+f_m$.
My question: From this post I know that in general $z$ is not in the convex hull of $x_1,...,x_m$. Can we impose certain conditions on $f_i$ and $x_i$ so that $z$ is not too far away from the convex hull of $x_1,...,x_m$?