The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant
In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, e.g. concerning the Helly dimension or the Banach–Mazur distance.
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- T. Bonnesen and W. Fenchel, Theory of convex bodies, BCS Associates, Moscow, ID, 1987, Translated from the German and edited by L. Boron, C. Christenson and B. Smith.Google Scholar
- R. Brandenberg, Radii of convex bodies, Ph.D. thesis, Zentrum Mathematik, Technische Universität München, 2002.Google Scholar
- L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives, in Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 101–180.Google Scholar
- H. G. Eggleston, Sets of constant width, in Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958, pp. 122–135.Google Scholar
- B. Grünbaum, Measures of symmetry for convex sets, in Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 233–270.Google Scholar
- R. Schneider, Convex bodies: the Brunn-Minkowski theory, expanded ed., Encyclopedia of Mathematics and its Applications, Vol. 151, Cambridge University Press, Cambridge, 2014.Google Scholar