Narrowing the gaps of the missing Blaschke-Santal\'o diagrams

TL;DR

This paper establishes new sharp inequalities among geometric measures of convex bodies, filling gaps in Blaschke-Santaló diagrams and extending some results to higher dimensions.

Contribution

It introduces several new sharp inequalities relating area, perimeter, inradius, circumradius, diameter, and width, and extends some to higher dimensions.

Findings

Sharp inequalities for planar convex bodies.

Narrowed gaps in Blaschke-Santaló diagrams.

Extended inequalities to higher dimensions.

Abstract

We solve several new sharp inequalities relating three quantities amongst the area, perimeter, inradius, circumradius, diameter, and minimal width of planar convex bodies. As a consequence, we narrow the missing gaps in each of the missing planar Blaschke-Santal\'o diagrams. Furthermore, we extend some of those sharp inequalities into higher dimensions, by replacing either the perimeter by the mean width or the area by the volume.