Isoperimetric inequalities for the mixed area of plane convex sets (Q1204070)

Let \(A(K,M)\) denote the mixed area of plane convex bodies \(A\) and \(M\), and \(L(K)\) shall stand for the perimeter of \(K\). The authors show \[ A(K,M)\leq{1\over 8}L(K)L(M) \] with equality precisely for \(K\) and \(M\) being orthogonal segments or if one of both sets is a point. Further on, \[ A(K,-K)\leq\left({\sqrt 3\over 18}\right)L^ 2(K) \] is proved, where in the set of polygons the equality case is characteristic for \(K\) a regular triangle (or a point). This equality discussion for all compact convex subsets of \(E^ 2\) remains open. An interesting application of the second result is given, too: the authors give the minimal density of the Euler characteristic for stationary Boolean models in the plane. Concluding remarks refer to possible approaches to the higher-dimensional analogues.