A Pu-Bonnesen inequality

TL;DR

This paper establishes a Bonnesen-type inequality for the real projective plane, extending Pu's systolic inequality by incorporating geometric measures like circumradius and inradius, using advanced convex geometry tools.

Contribution

It introduces a new inequality for the real projective plane that generalizes Pu's systolic inequality, utilizing John ellipsoids and Pogorelov's rigidity theorem.

Findings

Proves a Bonnesen-type inequality for the real projective plane.

Relates the inequality's remainder term to R-r, normalized.

Utilizes convex geometry and rigidity theorems in the proof.

Abstract

We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu's systolic inequality for positively-curved metrics. The remainder term in the inequality, analogous to that in Bonnesen's inequality, is a function of R-r (suitably normalized), where R and r are respectively the circumradius and the inradius of the Weyl-Lewy Euclidean embedding of the orientable double cover. We exploit John ellipsoids of a convex body and Pogorelov's ridigity theorem.