Convex bodies with extremal volumes having prescribed brightness in finitely many directions (Q1900087)

Let \(K\) be a convex body in \(n\)-dimensional Euclidean space and let \(W = \{w_1, \dots, w_m\}\) be a set of unit vectors containing a basis of the space. The symbol \({\mathcal H} (K,W)\) denotes the family of convex bodies whose \((n - 1)\)-dimensional volumes of the projections onto hyperplanes orthogonal to \(w_1, \dots, w_m\) coincide with the volumes of analogical projections of \(K\). The authors prove that there exists a unique convex body (up to translation) in \({\mathcal H} (K,W)\) with maximum volume. This body is a centrally-symmetric polytope. Moreover, it is proved that every convex body from \({\mathcal H} (K,W)\) of the minimum volume is a polytope with at most \(n + m\) facets.