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$\text{SL}(n)$ Invariant Valuations on Super-Coercive Convex Functions

Part of: Topological linear spaces and related structures General convexity Functions of several variables Polytopes and polyhedra

Published online by Cambridge University Press:  25 October 2019

Fabian Mussnig
Fabian Mussnig*
Affiliation:
School of Mathematical Sciences, Tel Aviv University, 69978Tel Aviv, Israel Email: mussnig@gmail.com
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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.

Keywords

valuationconvex functionsuper-coerciveLegendre transformSL(n) invariancepolar volume

MSC classification

Primary: 26B25: Convexity, generalizations
Secondary: 46A40: Ordered topological linear spaces, vector lattices 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A41: Convex functions and convex programs 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 1 , February 2021 , pp. 108 - 130
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Copyright
© Canadian Mathematical Society 2019

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