A Zoo of Dualities

TL;DR

This paper explores order reversing quasi involutions, their properties, and constructions, providing new examples, dualities, and inequalities, thereby deepening understanding and opening new research directions in convexity and duality theory.

Contribution

It introduces a unified framework for order reversing quasi involutions, constructs new examples including dual polarity, and establishes a Blaschke-Santaló type inequality for Gaussian volume products.

Findings

Characterization of order reversing quasi involutions induced by a cost

Existence and uniqueness results for invariant sets and extensions

A new example of dual polarity with a related inequality

Abstract

In this note we study order reversing quasi involutions and their properties. These maps are dualities (order reversing involutions) on their image. We prove that any order reversing quasi involution is induced by a cost. Invariant sets of order reversing quasi involutions are of special interest and we provide several results regarding their existence and uniqueness. We determine when an order reversing quasi involution on a sub-class can be extended to the whole space and discuss the uniqueness of such an extension. We also provide several ways for constructing new order reversing quasi involutions from given ones. In particular, we define the dual of an order reversing quasi involution. Finally, throughout the paper we exhibit a "zoo" of illustrative examples. Some of them are classical, some have recently attracted attention of the convexity community and some are new. We study in depth the new example of dual polarity and obtain a Blaschke-Santal\'o type inequality for a corresponding Gaussian volume product. The unified point of view on order reversing quasi involutions presented in this paper gives a deeper understanding of the underlying principles and structures, offering a new and exciting perspective on the topic, exposing many new research directions.