A Zoo of Dualities

S. Artstein-Avidan, S. Sadovsky, K. Wyczesany*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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-Santaló 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.

Original languageEnglish
Article number238
JournalJournal of Geometric Analysis
Volume33
Issue number8
DOIs
StatePublished - Aug 2023

Funding

FundersFunder number
Horizon 2020 Framework Programme770127
Iowa Science Foundation784/20
European Research Council
Azrieli Foundation

    Keywords

    • Duality
    • Order reversing
    • Quasi involution
    • Santaló type inequalities

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