Steinitz theorem on convex hull interiors

Remarks

It is an interior analogue of Carathéodory's theorem, where $d + 1$ points suffice for membership in the hull but $2d$ points are needed for the interior.

The bound $2d$ is sharp. Let $S = \set{\pm e_1, \dots, \pm e_d}$ be the vertices of the cross-polytope and let $x = 0$. If $T$ omits, say, $e_i$, then every point of $\operatorname{conv}(T)$ has $i$-th coordinate at most $0$, so $0$ is not interior to $\operatorname{conv}(T)$.

Example: in the plane, a point inside the convex hull of a finite set lies inside some quadrilateral, or some triangle, with vertices in the set.

The interior is taken in $\R^d$, so the hypothesis already forces $\operatorname{conv}(S)$ to have dimension $d$.

Going deeper

For a student

The theorem was proved by Ernst Steinitz, who also extended Carathéodory's theorem from compact to arbitrary sets. It is a standard companion of Carathéodory's and Helly's theorems in combinatorial convexity. It is used for positively spanning sets. A finite set positively spans $\R^d$ exactly when $0$ is in the interior of its convex hull. By the theorem, a subset of at most $2d$ of its vectors then already positively spans $\R^d$.