On the Homotopy Type of Balanced subsets

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 $\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{t}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(S))\cap r\mathrm{\ne }\mathrm{\varnothing }$. In the case when $r$ is a point in the relative interior of the whole set $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(V)$, 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 $V$.