Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 30J / b / Solution

The convex hull of $S\subseteq {\mathbb{R}}^d$ isThus $v$ is a convex combination of $v_1,\dots ,v_m$ when it has precisely this form. Equivalently, $\mathrm{conv}S$ is the smallest convex set containing $S$. The Caratheodory theorem sharpens the finite-combination description in ${\mathbb{R}}^d$: every point of $\mathrm{conv}S$ is a convex combination of at most $d+1$ points of $S$.