$\text{SL}(n)$ Invariant Valuations on Super-Coercive Convex Functions. (25th February 2021)

• Record Type: Journal Article
• Authors: Mussnig, Fabian
• Is Part Of: Canadian journal of mathematics. Volume 73:Number 1(2021)
• Page Start: 108
• Page End: 130
• Publication Date: 2021-02-25
• Subjects: 26B25, -- 46A40, -- 52A20, -- 52A41, -- 52B45; valuation, -- convex function, -- super-coercive, -- Legendre transform, -- SL(n) invariance, -- polar volume; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X19000531
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: All non-negative, continuous, $\text{SL}(n)$, and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^{n}$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.