On the Homotopy Type of Balanced subsets

Abstract: For a finite set of points $V={v_1, \dots, v_m}$ in Euclidean space $\mathbb{R}^d$ and a point $r \in \mathbb{R}^d$, a subset $S \subset V$ is called $r$-balanced if $\mathrm{relint}(\mathrm{conv}(S)) \cap r \neq \emptyset$. In the case when $r$ is a point in the relative interior of the whole set $\mathrm{conv}(V)$, we prove that the poset of all balanced subsets, excluding the whole set $V$, is homotopy equivalent to the sphere of dimension $m-k-2$, where $k$ is the dimension of the affine hull of $V$.