An isoperimetric inequality for convex polygons and convex sets with the same symmetrals (Q1067216)

Let H and K be two plane compact convex bodies with the same symmetrals in n different directions (n\(\geq 3)\). By using new isoperimetric inequalities for convex polygons it is proved that \((*) meas (H\Delta K) \leq (\ell^ 2/8n)\tan \pi /n,\) where \(\ell\) denotes the length of \(\partial (H\cap K)\). Furthermore it is shown that equality holds in (*) if and only if the directions are equally spaced and H and K are two regular concentric polygons with n sides, each obtained from the other by rotation through an angle \(\pi\) /n.