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:



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> 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 $z\in K$.
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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.



Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.
The answer is YES.


The function $\mathrm{dist}\_x$ is strictly convex at any point $y$ and any direction different $x-y$.


It follows that $f=\sum \mathop{\rm dist}\_{x\_i}$ is strictly convex at any point if $x\_i$ do not lie on one line;
in this case uniquness is obvious.


If there is a line, say $\ell$ containing all $x\_i$, then $f$ is strictly convex in any half-space not intersecting $\ell$. Hence the result.


