Bonnesen-style symmetric mixed isoperimetric inequality

Abstract

For convex domains K i (i = 0, 1) (compact convex sets with nonempty interiors) in the Euclidean plane R 2. Denote by A i and P i areas and circum-perimeters, respectively. The symmetric mixed isoperimetric deficit is ∆(K 0;K 1):= P 2 0 P 2 1 - 16π 2 A 0 A 1. In this paper, we give some Bonnesen-style symmetric mixed inequalities, that is, inequalities of the form ∆(K 0;K 1) ≥BK 0;K 1, where BK 0;K 1 is a non-negative invariant of geometric significance and vanishes if and only if both K 0 and K 1 are discs. We also obtain some reverse Bonnesen-style symmetric mixed inequalities. Those inequalities are natural generalizations of known geometric inequalities, such as the known classical isoperimetric inequality.