The Rosenthal-Szasz inequality for normed planes

TL;DR

This paper extends the classical Rosenthal-Szasz inequality to normed planes, characterizing curves of constant width and providing bounds for perimeter using differential geometry methods.

Contribution

It generalizes the Rosenthal-Szasz inequality to arbitrary normed planes and characterizes curves of constant width using differential geometry techniques.

Findings

For Radon normed planes, an inequality analogous to the Euclidean case is established.

An upper bound for perimeter in the anti-norm is derived for arbitrary norms.

Characterization of curves of constant width in general normed planes is provided.

Abstract

We aim to study the classical Rosenthal-Szasz inequality for a plane whose geometry is given by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterization of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.