Let $X$ be a convex compact subset of $\mathbb{R}^d$. Prove that there exist a vector $u$ such that $-\frac{1}{d}X+u\subseteq X$.
My idea is that:
For each $x\in X$, put $A_x=\{u : -\frac{1}{d}x+ u\in X\}$. It is very easy to see that all of $A_x$ are convex and compact. If we can prove that each intersection of at most $d+1$ of them is non-empty, then by applying Helly theorem we will have $$\bigcap_{x\in X} A_x\neq\emptyset ,$$ now any point of the intersection satisfied in the desired condition.