Upper bounds for edge-antipodal and subequilateral polytopes
Tóm tắt
A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d / 2 + 1)
d
for any d ≥ 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I. Talata [19]. This is a constructive improvement to the result of A. Pór (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices.
Tài liệu tham khảo
A. Barvinok, A Course in Convexity, American Mathematical Society, Providence, RI, 2002.
K. Bezdek, T. Bisztriczky and K. Böröczky, Edge-antipodal 3-polytopes, Discrete and Computational Geometry (J. E. Goodman, J. Pach, and E. Welzl, eds.), MSRI Special Programs, Cambridge University Press, 2005.
T. Bisztriczky and K. Böröczky, On antipodal 3-polytopes, Rev. Roumaine Math. Pures Appl., 50 (2005), 477–481.
H. Busemann, Intrinsic area, Ann. Math., 48 (1947), 234–267.
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Heidelberg, 1988.
B. Csikós, Edge-antipodal convex polytopes — a proof of Talata’s conjecture, Discrete Geometry, Monogr. Textbooks Pure Appl. Math., vol. 253, Dekker, New York, 2003, 201–205.
L. Danzer and B. Grünbaum, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, Math. Z., 79 (1962), 95–99.
Z. Füredi, J. C. Lagarias and F. Morgan, Singularities of minimal surfaces and networks and related extremal problems in Minkowski space, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Providence, RI, 1991, 95–109.
B. Grunbaum, Convex polytopes, 2nd ed., Springer, New York, 2003.
V. Klee, Unsolved problems in intuitive geometry, Mimeographed notes, Seattle, 1960.
G. Lawlor and F. Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific J. Math., 166 (1994), 55–83.
H. Martini and V. Soltan, Antipodality properties of finite sets in Euclidean space, Discrete Math., 290 (2005), 221–228.
M. S. Mel’nikov, Dependence of volume and diameter of sets in an n-dimensional Banach space (in Russian), Uspehi Mat. Nauk, 18 (1963), 165–170.
C. M. Petty, Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc., 29 (1971), 369–374.
A. Pór, On e-antipodal polytopes, submitted.
P. S. Soltan, Analogues of regular simplexes in normed spaces (in Russian), Dokl. Akad. Nauk SSSR, 222 (1975), 1303–1305. English translation: Soviet Math. Dokl., 16 (1975), 787–789.
K. J. Swanepoel, Cardinalities of k-distance sets in Minkowski spaces, Discrete Math., 197/198 (1999), 759–767.
A. C. Thompson, Minkowski Geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996.
I. Talata, On extensive subsets of convex bodies, Period. Math. Hungar., 38 (1999), 231–246.