The ratios of diameter and width of reduced and of complete convex bodies in Minkowski spaces

Original Paper


The breadth of a convex body K within a normed space \({\mathbb {R}}^d\) in direction of a given hyperplane H is the distance between the two supporting hyperplanes of K parallel to H, measured in the underlying norm. The minimal and the maximal value over all directions are called minimal width and diameter of K. The body K is reduced if every body \(K_*\subsetneq K\) has a smaller minimal width, and K is complete if every body \(K^*\supsetneq K\) has a larger diameter. We study the ratio between diameter and minimal width of reduced and of complete bodies. If \(d \ge 3\) then there exist reduced bodies of arbitrarily large ratio, whereas the ratio for complete bodies is bounded by \(\frac{d+1}{2}\). As a consequence, every normed space \({\mathbb {R}}^d\), \(d \ge 2\), contains reduced bodies that are not complete.


Minkowski space Convex body Reduced Complete Minimal width Diameter 

Mathematics Subject Classification

52A20 52A21 52A40 


  1. Chakerian, G.D., Groemer, H.: Convex bodies of constant width. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its Applications, pp. 49–96. Birkhäuser, Basel (1983)CrossRefGoogle Scholar
  2. Eggleston, H.G.: Sets of constant width in finite dimensional Banach spaces. Israel J. Math. 3, 163–172 (1965)MathSciNetCrossRefMATHGoogle Scholar
  3. Fabińska, E., Lassak, M.: Reduced bodies in normed planes. Israel J. Math. 161, 75–87 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. Heil, E., Martini, H.: Special convex bodies. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 347–385. North-Holland, Amsterdam (1993)CrossRefGoogle Scholar
  5. Jessen, B.: Über konvexe Punktmengen konstanter Breite (On convex point sets of constant width, in German). Math. Z. 29, 378–380 (1929)MathSciNetCrossRefGoogle Scholar
  6. Kapacë Open image in new window, P.H. (Karasëv, R.N.): O Open image in new window (On the characterization of generating sets, in Russian). Modelirovaniye i Analiz Informatsionnyh Sistem 8, 3–9 (2001)Google Scholar
  7. Lassak, M., Martini, H.: Reduced bodies in Minkowski space. Acta Math. Hung. 106, 17–26 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. Lassak, M., Martini, H.: Reduced convex bodies in finite dimensional normed spaces: a survey. Results Math. 66, 405–426 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. Leichtweiss, K.: Zwei Extremalprobleme der Minkowski-Geometrie. (Two extremum problems of Minkowski geometry, in German). Math. Z. 62, 37–49 (1955)MathSciNetCrossRefMATHGoogle Scholar
  10. Martini, H., Swanepoel, K.: The geometry of Minkowski spaces—a survey. Part II. Expo. Math. 22, 93–144 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. Martini, H., Wu, S.: Complete sets need not be reduced in Minkowski spaces. Beitr. Algebra Geom. 56, 533–539 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. Moreno, J.P., Schneider, R.: Diametrically complete sets in Minkowski spaces. Israel J. Math. 191, 701–720 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Managing Editors 2017

Authors and Affiliations

  1. 1.Institut für MathematikFriedrich-Schiller-Universität JenaJenaGermany

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