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

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 \(f_K\). The function \(f_K\) measures the average taxicab distance of the points in the plane from the focal set \(K\) by integration. The main area of the applications is the geometric tomography because \(f_K\) involves the coordinate X-rays' information as second order partial derivatives \cite{NV3}. We prove that the Hausdorff-convergence implies the convergence of the conic functions with respect to both the supremum-norm and the \(L_1\)-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 \(f_L\) is close to \(f_K\) then \(L\) must be close to an element \(K'\) such that \(f_{K}=f_{K'}\). Therefore \(K\) and \(K'\) 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 \(f_K\) is a point of lower semi-continuity for \(\Phi^{-1}\).