Covariograms generated by valuations

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Let ϕbe a real-valued valuation on the family of compact convex subsets of Rⁿ and let K be a convex body in Rⁿ. We introduce the ϕ-covariogram gₖ,ϕ of K as the function associating to each x ∈ Rⁿ the value ϕ(K ∩ (K+x)). If ϕis the volume, then gₖ,ϕ is the covariogram, extensively studied in various sources. When ϕis a quermassintegral (e.g., surface area or mean width) gₖ,ϕ has been introduced by Nagel. We study various properties of ϕ-covariograms, mostly in the case n=2 and under the assumption that ϕis translation invariant, monotone and even. We also consider the generalization of Matheron's covariogram problem to the case of ϕ-covariograms, that is, the problem of determining an unknown convex body K, up to translations and point reflections, by the knowledge of gₖ,ϕ. A positive solution to this problem is provided under different assumptions, including the case that K is a polygon and ϕis either strictly monotone or ϕis the width in a given direction. We prove that there are examples in every dimension n≥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.