Affinely embeddable convex bodies (Q1100729)

Let \({\mathcal B}=\{B_ 1,...,B_ n\}\) be a collection of \(n\geq 2\) convex sets in a real projective 3-space \(P^ 3\). If each \(B_ i\in {\mathcal B}\) is convex in the same affine restriction of \(P^ 3\) then \({\mathcal B}\) is called affinely embeddable. In the present article some conditions under which \({\mathcal B}\) is affinely embeddable are established. Theorem. Let \(n\geq 5\), \(B_ i\cap B_ j=\emptyset\) (i\(\neq j)\), \(B_ i\) closed. Assume that no two sets of \({\mathcal B}\) are coplanar, any line meets at most two sets of \({\mathcal B}\) and a set of \({\mathcal B}\) is neither a point nor contained in a convex hull of any other three sets of \({\mathcal B}\). Then \({\mathcal B}\) is affinely embeddable.