Stochastic reverse isoperimetric inequalities in the plane

TL;DR

This paper enhances classical planar isoperimetric inequalities by incorporating stochastic models, providing new dual inequalities and strengthening existing geometric bounds through probabilistic methods.

Contribution

It introduces stochastic models to strengthen classical inequalities like Mahler's Theorem and reverse Lutwak-Zhang inequality in the plane, including dual results.

Findings

Strengthened planar isoperimetric inequalities using stochastic models

Established dual counterparts to existing inequalities

Connected stochastic dominance with classical geometric inequalities

Abstract

In recent years, it has been shown that some classical inequalities follow from a local stochastic dominance for naturally associated random polytopes. We strengthen planar isoperimetric inequalities by attaching a stochastic model to some classical inequalities, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, the polar for $L_p$ centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-B\"or\"oczky.