Complexes of C-Unbalanced Subsets and Balanced-Subset Posets

Let \(V=\{v_1,\dots,v_m\}\subset\mathbb{R}^d\). We study simplicial complexes arising from subsets of \(V\) whose convex hulls avoid a prescribed convex set. Given a convex set \(C\subset\mathbb{R}^d\), let \(\mathcal K(V,C)\) be the simplicial complex consisting of all subsets \(S\subset V\) such t