Inner and outerj-radii of convex bodies in finite-dimensional normed spaces

This paper is concerned with the various inner and outer radii of a convex bodyC in ad-dimensional normed space. The innerj-radiusrj(C) is the radius of a largestj-ball contained inC, and the outerj-radiusRj(C) measures how wellC can be approximated, in a minimax sense, by a (d —j)-flat. In particular,rd(C) andRd(C) are the usual inradius and circumradius ofC, while 2r1(C) and 2R1(C) areC's diameter and width.Motivation for the computation of polytope radii has arisen from problems in computer science and mathematical programming. The radii of polytopes are studied in [GK1] and [GK2] from the viewpoint of the theory of computational complexity. This present paper establishes the basic geometric and algebraic properties of radii that are needed in that study.

[1]  A. Heppes.  Beweis einer Vermutung von A. Vázsonyi , 1956 .

[2]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[3]  G. C. Shephard,et al.  The difference body of a convex body , 1957 .

[4]  M. Day,et al.  Some characterizations of inner-product spaces , 1947 .

[5]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

[6]  A. L. Brown.  Best n‐Dimensional Approximation to Sets of Functions , 1964 .

[7]  H. G. Eggleston.  Notes on Minkowski Geometry (I): Relations between the Circumradius, Diameter, Inradius and Minimal Width of a Convex Set , 1958 .

[8]  Wilhelm Blaschke.  Kreis und Kugel , 1916 .

[9]  Andrew Chi-Chih Yao,et al.  On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems , 1977, SIAM J. Comput..

[10]  B. B. Panda,et al.  On equidistant sets in normed linear spaces , 1974, Bulletin of the Australian Mathematical Society.

[11]  Victor Klee,et al.  Circumspheres and Inner Products. , 1960 .

[12]  Peter Gritzmann,et al.  Good and Bad Radii of Convex Polygons , 1991, SIAM J. Comput..

[13]  M. J. Pelling.  Regular Simplices with Rational Vertices , 1977 .

[14]  Heinz Bauer,et al.  Minimalstellen von Funktionen und Extremalpunkte , 1958 .

[15]  Jan van Leeuwen,et al.  Computational complexity of norm-maximization , 1990, Comb..

[16]  Heinrich W. E. Jung.  Ueber die kleinste Kugel, die eine räumliche Figur einschliesst. , 1901 .

[17]  G. Lorentz.  Approximation of Functions , 1966 .

[18]  Shizuo Kakutani,et al.  Some characterizations of Euclidean space , 1940 .

[19]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[20]  I. Singer.  Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces , 1970 .

[21]  F. Bohnenblust,et al.  Convex Regions and Projections in Minkowski Spaces , 1938 .

[22]  A. Pinkus.  n-Widths in Approximation Theory , 1985 .

[23]  Peter Gritzmann,et al.  Estimates for the minimal width of polytopes inscribed in convex bodies , 1989, Discret. Comput. Geom..

[24]  Peter Gritzmann,et al.  Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces , 1993, Math. Program..

[25]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[26]  V. Tikhomirov,et al.  DIAMETERS OF SETS IN FUNCTION SPACES AND THE THEORY OF BEST APPROXIMATIONS , 1960 .

[27]  I. J. Schoenberg.  Regular Simplices and Quadratic Forms , 1937 .

[28]  Paul Steinhagen,et al.  Über die größte Kugel in einer konvexen Punktmenge , 1922 .

[29]  T. Bonnesen,et al.  Theorie der Konvexen Körper , 1934 .

[30]  Victor Klee,et al.  Inspheres and inner products , 1986 .

[31]  A. Kolmogoroff,et al.  Uber Die Beste Annaherung Von Funktionen Einer Gegebenen Funktionenklasse , 1936 .

[32]  Kurt Leichtweiss.  Zwei Extremalprobleme der Minkowski-Geometrie , 1955 .