Covariograms generated by valuations

Let phi be a real-valued valuation on the family of n-dimensional compact convex subsets of R^n and let K be a convex body in R^n. We introduce the phi-covariogram g_{K,phi} of K as the function associating to each x in 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 situations in every dimension n geq 3 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.