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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Generalized Hadwiger numbers for symmetric ovals
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by Valentin Boju and Louis Funar PDF
Proc. Amer. Math. Soc. 119 (1993), 931-934 Request permission

Abstract:

Some estimations for the "juxtaposition function" ${h_F}$ and an asymptotic formula for the function ${h_F}/{h_G}$, where $F,\;G$ are central symmetric convex bodies, are given. Hadwiger and Grünbaum gave for ${h_F}(1)$ the bounds ${n^2} + n \leqslant {h_F}(1) \leqslant {3^n} - 1$. Grünbaum conjectured (and proved for $n = 2$ in Pacific J. Math. 11 (1961), 215-219) that for every even $r$ between these bounds there exists in ${E^n}$ an oval $F$ such that ${h_F}(1) = r$. Lower bounds for ${h_F}$ could be derived in the same way as in Theorems 1 and 2 from a good estimate of packing numbers on a Minkowski sphere, that is, from solutions to a Tammes-type problem in a Banch space.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 119 (1993), 931-934
  • MSC: Primary 52C15; Secondary 52A10, 52C17
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1176065-5
  • MathSciNet review: 1176065