Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays
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| 024 | 7 | 0 | |a 10.1007/s00010-014-0322-2 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0322-2 | ||
| 245 | 0 | 0 | |a Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays |h [Elektronische Daten] |c [Csaba Vincze, Ábris Nagy] |
| 520 | 3 | |a 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 . | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Hausdorff metric |2 nationallicence | |
| 690 | 7 | |a parallel X-ray |2 nationallicence | |
| 690 | 7 | |a set-valued mapping |2 nationallicence | |
| 690 | 7 | |a Generalized conic function |2 nationallicence | |
| 700 | 1 | |a Vincze |D Csaba |u Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary |4 aut | |
| 700 | 1 | |a Nagy |D Ábris |u Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/4(2015-08-01), 1015-1030 |x 0001-9054 |q 89:4<1015 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0322-2 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0322-2 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Vincze |D Csaba |u Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Nagy |D Ábris |u Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/4(2015-08-01), 1015-1030 |x 0001-9054 |q 89:4<1015 |1 2015 |2 89 |o 10 | ||