The Lower Bounds of the Mixed Isoperimetric Deficit

The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

“…The wellknown Blaschke-Süss theorem states that there are at least three pairs of antipodal points on an oval ( [22,26]). The absolute value of the oriented area of the Wigner caustic gives the exact relation between the perimeter and the area of the region bounded by closed regular curves of constant width and improves the classical isoperimetric inequality for convex curves ( [34,35,[37][38][39]). Furthermore this oriented area improves the isoperimetric defect in the reverse isoperimetric inequality ( [7]).…”