Author:
Károly Bezdek Department of Mathematics and Statistics, University of Calgary, Canada
Department of Mathematics, University of Pannonia, Veszprém, Hungary

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Let 𝔼𝑑 denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼𝑑 is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 < 𝑘 < 𝑑 and for any set of given 𝑑-dimensional volume in 𝔼𝑑 the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.

  • [1]

    Bezdek, K., Lángi, Zs., Naszódi, M., and Papez, P. Ball-polyhedra. Discrete Comput. Geomg. 38, 2 (2007), 201230.

  • [2]

    Bezdek, K. From 𝑟-dual sets to uniform contractions. Aequationes Math. 92, 1 (2018), 123134.

  • [3]

    Bezdek, K. and Naszódi, M. The Kneser–Poulsen conjecture for special contractions. Discrete Comput. Geom. 60, 4 (2018), 967980.

  • [4]

    Bezdek, K. On the intrinsic volumes of intersections of congruent balls. Discrete Optim. 44, 1 (2022), Paper No. 100539 (7 pages).

  • [5]

    Borisenko, A. A. and Drach, K. D. Isoperimetric inequality for curves with curvature bounded below. Matematicheskie Mat. Zametki 95, 5 (2014), 656665; English translation: Math. Notes 95, 5–6 (2014), 590–598.

    • Search Google Scholar
    • Export Citation
  • [6]

    Fejes Tóth, L. Packing of 𝑟-convex discs. Studia Sci. Math. Hungar. 17, 1–4 (1982), 449452.

  • [7]

    Fodor, F., Kurusa, Á., and Vígh, V. Inequalities for hyperconvex sets. Adv. Geom. 16, 3 (2016), 337348.

  • [8]

    Gao, F., Hug, D., and Schneider, R. Intrinsic volumes and polar sets in spherical space. Math. Notae 41 (2003), 159176.

  • [9]

    Gardner, R. J. The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 3 (2002), 355405.

  • [10]

    Kupitz, Y. S., Martini, H., and Perles, M. A. Ball polytopes and the Vázsonyi problem. Acta Math. Hungar. 126, 1–2 (2010), 99163.

  • [11]

    Lángi, Zs., Naszódi, M., and Talata, I. Ball and spindle convexity with respect to a convex body. Aequationes Math. 85, 1–2 (2013), 4167.

    • Search Google Scholar
    • Export Citation
  • [12]

    Mayer, A. E. Eine Überkonvexität. Math. Z. 39, 1 (1935), 511531.

  • [13]

    Paouris, G. and Pivovarov, P. Random ball-polyhedra and inequalities for intrinsic volumes. Monatsh. Math. 182, 3 (2017), 709729.

  • [14]

    Schneider, R. Convex bodies: the Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press, Cambridge, 1993.

    • Search Google Scholar
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Editor in Chief: László TÓTH (University of Pécs)

Honorary Editors in Chief:

  • † István GYŐRI (University of Pannonia, Veszprém, Hungary)
  • János PINTZ (Rényi Institute of Mathematics, Budapest, Hungary)
  • Ferenc SCHIPP (Eötvös Loránd University, Budapest, Hungary and University of Pécs, Pécs, Hungary)
  • Sándor SZABÓ (University of Pécs, Pécs, Hungary)
     

Deputy Editors in Chief:

  • Erhard AICHINGER (JKU Linz, Linz, Austria)
  • Ferenc HARTUNG (University of Pannonia, Veszprém, Hungary)
  • Ferenc WEISZ (Eötvös Loránd University, Budapest, Hungary)

Editorial Board

  • Attila BÉRCZES (University of Debrecen, Debrecen, Hungary)
  • István BERKES (Rényi Institute of Mathematics, Budapest, Hungary)
  • Károly BEZDEK (University of Calgary, Calgary, Canada)
  • György DÓSA (University of Pannonia, Veszprém, Hungary)
  • Balázs KIRÁLY – Managing Editor (University of Pécs, Pécs, Hungary)
  • Vedran KRCADINAC (University of Zagreb, Zagreb, Croatia) 
  • Željka MILIN ŠIPUŠ (University of Zagreb, Zagreb, Croatia)
  • Gábor NYUL (University of Debrecen, Debrecen, Hungary)
  • Margit PAP (University of Pécs, Pécs, Hungary)
  • István PINK (University of Debrecen, Debrecen, Hungary)
  • Mihály PITUK (University of Pannonia, Veszprém, Hungary)
  • Lukas SPIEGELHOFER (Montanuniversität Leoben, Leoben, Austria)
  • Andrea ŠVOB (University of Rijeka, Rijeka, Croatia)
  • Csaba SZÁNTÓ (Babeş-Bolyai University, Cluj-Napoca, Romania)
  • Jörg THUSWALDNER (Montanuniversität Leoben, Leoben, Austria)
  • Zsolt TUZA (University of Pannonia, Veszprém, Hungary)

Advisory Board

  • Szilárd RÉVÉSZ (Rényi Institute of Mathematics, Budapest, Hungary)  - Chair
  • Gabriella BÖHM
  • György GÁT (University of Debrecen, Debrecen, Hungary)

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Mathematica Pannonica
Language English
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ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)