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

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• 作者：Csaba Vincze ; ábris Nagy
• 关键词：26B15 ; 26B25 ; Hausdorff metric ; parallel X ; ray ; set ; valued mapping ; Generalized conic function
• 刊名：Aequationes Mathematicae
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：89
• 期：4
• 页码：1015-1030
• 全文大小：553 KB
• 参考文献：1.Coppel W.A.: Foundations of convex geometry. Cambridge University Press, Cambridge (1998)MATH
  2.Gardner R.J.: Geometric tomography. Cambridge University Press, Cambridge (1995)MATH
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  5.Schneider R.: Convex bodies: the Brunn–Minkowski theory. Cambridge University Press, Cambridge (1993)MATH View Article
  6.Nagy á., Vincze Cs.: Examples and notes on generalized conics and their applications. AMAPN 26(2), 359-75 (2010)MATH MathSciNet
  7.Nagy á., Vincze Cs.: An introduction to the theory of generalized conics and their applications. J. Geom. Phys. 61(4), 815-28 (2011)MATH MathSciNet View Article
  8.Nagy á., Vincze Cs.: On the theory of generalized conics with applications in geometric tomography. J. Approx. Theory 164, 371-90 (2012)MATH MathSciNet View Article
  9.Nagy, á., Vincze, Cs.: Reconstruction of hv-convex sets by their coordinate X-ray functions. J. Math. Imaging Vis. 49(3), 569-82 (2014)
  
• 作者单位：Csaba Vincze (1)
  ábris Nagy (1) (2) (3)
  
  1. Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary
  2. Institute of Mathematics, MTA-DE Research Group ‘Equations Functions and Curves- Debrecen, Hungary
  3. Hungarian Academy of Sciences, Debrecen, Hungary
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Combinatorics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8903

文摘

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 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: 371-90, 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 \({\Phi^{-1}}\) 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 \({K^\prime}\) 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 f K is a point of lower semi-continuity for \({\Phi^{-1}}\).