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

TL;DR

This paper classifies $ m{SL}(n)$ covariant function-valued valuations under continuity assumptions, introducing new valuations like weighted moment functions and unifying various transforms and bodies in convex geometry.

Contribution

It provides a comprehensive classification of $ m{SL}(n)$ covariant valuations, introduces new valuation types, and unifies several important convex geometric transforms and bodies.

Findings

Classified $ m{SL}(n)$ covariant function-valued valuations with continuity.

Introduced new valuations such as weighted moment functions.

Unified characterizations of Laplace transform, $L_p$ bodies, and polar $L_p$ bodies.

Abstract

Classifications of $\rm{SL}(n)$ 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.