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.
Bezdek, K., Lángi, Zs., Naszódi, M., and Papez, P. Ball-polyhedra. Discrete Comput. Geomg. 38, 2 (2007), 201–230.
Bezdek, K. From 𝑟-dual sets to uniform contractions. Aequationes Math. 92, 1 (2018), 123–134.
Bezdek, K. and Naszódi, M. The Kneser–Poulsen conjecture for special contractions. Discrete Comput. Geom. 60, 4 (2018), 967–980.
Bezdek, K. On the intrinsic volumes of intersections of congruent balls. Discrete Optim. 44, 1 (2022), Paper No. 100539 (7 pages).
Borisenko, A. A. and Drach, K. D. Isoperimetric inequality for curves with curvature bounded below. Matematicheskie Mat. Zametki 95, 5 (2014), 656–665; English translation: Math. Notes 95, 5–6 (2014), 590–598.
Fejes Tóth, L. Packing of 𝑟-convex discs. Studia Sci. Math. Hungar. 17, 1–4 (1982), 449–452.
Fodor, F., Kurusa, Á., and Vígh, V. Inequalities for hyperconvex sets. Adv. Geom. 16, 3 (2016), 337–348.
Gao, F., Hug, D., and Schneider, R. Intrinsic volumes and polar sets in spherical space. Math. Notae 41 (2003), 159–176.
Gardner, R. J. The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 3 (2002), 355–405.
Kupitz, Y. S., Martini, H., and Perles, M. A. Ball polytopes and the Vázsonyi problem. Acta Math. Hungar. 126, 1–2 (2010), 99–163.
Lángi, Zs., Naszódi, M., and Talata, I. Ball and spindle convexity with respect to a convex body. Aequationes Math. 85, 1–2 (2013), 41–67.
Mayer, A. E. Eine Überkonvexität. Math. Z. 39, 1 (1935), 511–531.
Paouris, G. and Pivovarov, P. Random ball-polyhedra and inequalities for intrinsic volumes. Monatsh. Math. 182, 3 (2017), 709–729.
Schneider, R. Convex bodies: the Brunn–Minkowski theory. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press, Cambridge, 1993.