$\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations

Abstract

Classifications of $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.

References in corpus (11)

• The Logarithmic Minkowski Problem
• Crofton measures and Minkowski valuations
• General Lp affine isoperimetric inequalities
• Minkowski valuations on convex functions
• A homogeneous decomposition theorem for valuations on convex functions
• Laplace transforms and valuations
• Volume, Polar Volume and Euler Characteristic for Convex Functions
• $L_p$ Minkowski Valuations on polytopes
• Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem
• Affine invariant valuations on polytopes
• Affine function valued valuations

Cited by in corpus (3)

• Geometric valuation theory
• $\mathrm{SL}(n)$ contravariant function-valued valuations on polytopes
• The Legendre transform, the Laplace transform and valuations