If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the quantity $\|z-x_0\|$ in $z\in K$.

My question is whether it is known if the obvious generalization holds:

Given two (or n) points $x_0,x_1\notin K$ so that the set $\{x_0,x_1\}$ is separated from $K$ by an affine hyperplane, then there exists a unique point $z_0\in K$ minimizing the quantity $\|z-x_0\|+\|z-x_1\|$ in $z\in K$.

My feeling is that this should have been investigated somewhere in the literature, but in my inexperience navigating that landscape I had thus far been unable to find it.

Geometry of geodesics). In the higher dimensional case (or the general Banach space case) I have not seen the sets $\|z -x_0\| + \|z -x_1\| \leq d$ studied. It is easy to see that these (sort of) "ellipsoids" are convex, so I guess your question reduces to something like:if the unit ball of a Banach space is strictly convex, is the same true for all "ellipsoids"?$\endgroup$