Is the center of the smallest enclosing circle in convex hull?

Question

Let suppose you have a finite set of points $x_1, \ldots, x_n \in \mathbb{R}^d$.

Let $x_c = \underset{y \in \mathbb{R}^d}{\arg\min}~\underset{i = 1, \ldots, n}{\max} \| x_i - y \|_2^2$ be the center of the smallest enclosing circle of $x_1, \ldots, x_n$ (https://en.wikipedia.org/wiki/Smallest-circle_problem)

My question is the following :

Is $x_c$ necessarily a convex combination of $x_1, \ldots, x_n$ ?

Answer 1

If the centre is not within the convex hull, there's a plane separating the centre from the points. Move the centre towards that plane, and you've moved it closer to all the points; a contradiction.