Covariograms generated by valuations

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.