Let $K$ be an $o$-symmetric body, that covers $X$. Does it imply $\operatorname{conv}(X)\subset X+K$?

Question

The question is inspired by the answer (and a comment) of daw at Covering of a polytope by balls with midpoints at the vertices

Let $K\subset \mathbb R^n$ be an $o$-symmetric ($K=-K$) convex body. Does $X\subset K$ imply $\operatorname{conv}(X)\subset X+K$? We already know, that it's true, if $K$ is an euclidean ball or ellipsoid.

Answer 1

Here is a counterexample. Take $K$ to be the unit $1$-ball in $\Bbb{R}^3$: $$K = \{(x, y, z) : |x| + |y| + |z| \le 1\},$$ and $$X = \{e_1, e_2, e_3\} = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\},$$ respectively. Note that $x \in X + K$ if and only if $\|x - e_i\|_1 \le 1$ for $i = 1, 2, 3$.

If we consider the point $x = (1/3, 1/3, 1/3)$, then we see that $\|x - e_i\| = \frac{4}{3} > 1$ for all $i$, so $x \in \operatorname{conv} X \setminus (X + K)$, proving the conjecture false.