Strict dominance wrt distances to vertices in a convex hull

Question

We have a convex hull of some points $V=conv (x_1,\cdots,x_n)$. Is it true that no point in $V$ strictly dominates another point in $V$ with respect to distances to the verices? I mean that $\forall y,y'\in V $ it doesnt hold that $||x_i-y||>||x_i-y'||,\ i=1,\cdots,n$. How to prove it?

Answer 1

Hint: Here is a geometric idea, you can adopt it to a rigors analytic proof.

Let $z$ be the midpoint of the segment joining $y$ to $y' $. Consider the hyper plan cutting this segment at point $z$. i.e., the hyper plane with the normal vector $n= y- y'$ passing through $z$. This hyperplane divides the set $V$ into two nontrivial DISJOINT parts $A$ , $B$ such that $ y \in A$ and $y' \in B$. Actually $A = V \cap H^+$ and $B = V \cap H^-$ where $H^+$ and $H^-$ are two open half space of the hyperplan $H.$ It is not hard to show that $A$ contains a vertex of $V$, say $x \in A$ is a vertex of $V$ then it is clear that

$$||x-y'||>||x-y||$$

Contradiction.