Planar rectangular sets and Steiner symmetrization (Q1267723)

The paper concerns (in fact) compact convex subsets of the Euclidean plane (though the author does not assume compactness). A set \(K\) is said to be rectangular with respect to a pair of orthogonal lines if no inscribed rectangle with sides parallel to those lines has exactly three of its vertices on the boundary of \(K\). Let \(S_L\) be the Steiner symmetrization with respect to a line \(L\). Theorem. \(K\) is rectangular with respect to \((L_1,L_2)\) if and only if \(S_{L_1} S_{L_2}(K)= S_{L_2} S_{L_1}(K)\).