Covariograms generated by valuations

TL;DR

This paper explores generalized covariograms derived from valuations on convex bodies, analyzing their properties, and solving the inverse problem of reconstructing bodies from these covariograms under various conditions.

Contribution

It introduces the concept of ovariograms for valuations, studies their properties in low dimensions, and provides conditions for unique determination of convex bodies from these covariograms.

Findings

ovariograms extend classical covariogram concepts.

Unique determination of convex bodies is possible under certain valuation conditions.

Counterexamples show non-uniqueness in higher dimensions.

Abstract

Let \phi be a real-valued valuation on the family of compact convex subsets of \mathbb{R}^n and let K be a convex body in \mathbb{R}^n. We introduce the \phi -covariogram g_{K,\phi} of K as the function associating to each x \in \mathbb{R}^n the value \phi(K \cap (K+x)). If \phi is the volume, then g_{K,\phi} is the covariogram, extensively studied in various sources. When \phi is a quermassintegral (e.g., surface area or mean width) g_{K,\phi} has been introduced by Nagel. We study various properties of \phi -covariograms, mostly in the case n=2 and under the assumption that \phi is translation invariant, monotone and even. We also consider the generalization of Matheron's covariogram problem to the case of \phi -covariograms, that is, the problem of determining an unknown convex body K, up to translations and point reflections, by the knowledge of g_{K,\phi}. A positive solution to this problem is provided under different assumptions, including the case that K is a polygon and \phi is either strictly monotone or \phi is the width in a given direction. We prove that there are examples in every dimension n\geq3 where K is determined by its covariogram but it is not determined by its width-covariogram. We also present some consequence of this study in stochastic geometry.