Affinely embeddable separated families (Q2639339)

Let \(P^ d\) denote the real projective d-space, \(d\geq 2\). A subset B of \(P^ d\) is convex if it is disjoint from some hyperplane \(\alpha\) (bounded) and convex in the affine restriction \(P^ d\setminus \alpha\). A family \({\mathcal B}\) of convex sets in \(P^ d\) is called affinely embeddable if each \(B\in {\mathcal B}\) is convex in the same restriction of \(P^ d\). The family \({\mathcal B}\) is said to be separated in \(P^ d\) if any m-flat of \(P^ d\) \((0\leq m\leq d-2)\) meets at most \(m+1\) members of \({\mathcal B}\). Theorem: Let \({\mathcal B}=\{B_ 1,B_ 2,...,B_ n\}\) be a separated family of \(n\geq d+2\) closed smooth convex bodies in \(P^ d\) such that no member of \({\mathcal B}\) is contained in a convex hull of any other d members of \({\mathcal B}\), \(d\geq 2\). Then \({\mathcal B}\) is affinely embeddable.