caa a22 4500 60550878X CHVBK 20210128100638.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s00010-014-0322-2 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0322-2 Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays [Elektronische Daten] [Csaba Vincze, Ábris Nagy] In the paper we investigate the continuity properties of the mapping $${\Phi}$$ Φ 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 applications is geometric tomography because f K 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 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 $${\Phi^{-1}}$$ Φ - 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 $${K^\prime}$$ K ′ such that $${f_{K}=f_{K^\prime}}$$ f K = f K ′ . Therefore K and $${K^\prime}$$ 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 $${\Phi^{-1}}$$ Φ - 1 . Springer Basel, 2014 Hausdorff metric nationallicence parallel X-ray nationallicence set-valued mapping nationallicence Generalized conic function nationallicence Vincze Csaba Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut Nagy Ábris Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut Aequationes mathematicae Springer Basel 89/4(2015-08-01), 1015-1030 0001-9054 89:4<1015 2015 89 10 https://doi.org/10.1007/s00010-014-0322-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0322-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Vincze Csaba Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut NATIONALLICENCE 700 1- Nagy Ábris Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/4(2015-08-01), 1015-1030 0001-9054 89:4<1015 2015 89 10