Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

Abstract

In the paper we investigate the continuity properties of the mapping \({\Phi}\) which sends any non-empty compact connected hv-convex planar set K to the associated generalized conic function fK. The function fK measures the average taxicab distance of the points in the plane from the focal set K by integration. The main area of applications is geometric tomography because fK involves the coordinate X-rays’ information as second order partial derivatives (Nagy and Vincze, J Approx Theory 164: 371–390, 2012). We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the L1-norm provided that we restrict the domain to the collection of non-empty compact connected hv-convex planar sets contained in a fixed box (reference set) with parallel sides to the coordinate axes. We also have that \({\Phi^{-1}}\) is upper semi-continuous as a set-valued mapping. The upper semi-continuity establishes an approximating process in the sense that if fL is close to fK then L must be close to an element \({K^\prime}\) such that \({f_{K}=f_{K^\prime}}\). Therefore K and \({K^\prime}\) have the same coordinate X-rays almost everywhere. Lower semi-continuity is usually related to the existence of continuous selections. If a set-valued mapping is both upper and lower semi-continuous at a point of its domain it is called continuous. The last section of the paper is devoted to the case of non-empty compact convex planar sets. We show that the class of convex bodies that are determined by their coordinate X-rays coincides with the family of convex bodies K for which fK is a point of lower semi-continuity for \({\Phi^{-1}}\).