Pairs of convex bodies with centrally symmetric intersections of translates (Q2484003)

The paper under review gives a nice characterisation of pairs of convex bodies \(K_1,K_2\subset \mathbb{R}^n\) for which the \(n\)-dimensional intersections \(K_1\cap (x+K_2)\), \(x\in\mathbb{R}^n\), are centrally symmetric. The main result is: \(K_1,K_2\) have this property if and only if they admit a representation as direct sums \(K_i=R_i\oplus P_i\), such that i) \(R_1\) is a convex body of dimension \(m\in\{0,\dots,n\}\), and \(R_2\) is a translate of \(-R_1\) and ii) \(P_1\) and \(P_2\) are isothetic parallelotopes of dimension \(n-m\). Here two \((n-m)\)-dimensional parallelotopes are called isothetic if they can be represented as discrete sums \(P_i=l_i^1\oplus \dots\oplus l_i^{m-n}\) where \(l_1^j\) and \(l_2^j\) are parallel line segments contained in some subspaces \(L^j\) which also form a direct sum.