On a Problem by J. M. Wills

✍ Scribed by Peter Sauer

• Publisher: John Wiley and Sons
• Year: 1993
• Tongue: English
• Weight: 173 KB
• Volume: 162
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19931620118

✦ Synopsis

Abstract

Let P~1~(n), P~2~(n), P~3~(n) denote the norm‐polyhedra cube, crosspolytope and sphere in the Euclidean n‐space. We consider the polyhedra D~i,j~(n):=P~i~(n)∩P~j~(n) with 1 ≦ i ≠ j ≦ 3. In [2] posed the question about the limits \documentclass{article}\pagestyle{empty}\begin{document}$\mathop {\lim }\limits_{n \to \infty } $\end{document} V(D~i,j~(n)) of the volumes of the polyhedra D~i,j~(n). WEISSBACH gave the answer for one of them, namely \documentclass{article}\pagestyle{empty}\begin{document}$\mathop {\lim }\limits_{n \to \infty } $\end{document} V(D~1,2~(n)). A more general result used Banach‐space methods of Schechtman and ZINN contains the solution for the limits \documentclass{article}\pagestyle{empty}\begin{document}$\mathop {\lim }\limits_{n \to \infty } $\end{document} V(D~i,j~(n)). The main result of this paper is the determination of both limits \documentclass{article}\pagestyle{empty}\begin{document}$\mathop {\lim }\limits_{n \to \infty } $\end{document} V(D~1,3~(n)) and \documentclass{article}\pagestyle{empty}\begin{document}$\mathop {\lim }\limits_{n \to \infty } $\end{document} V(D~2,3~(n)) with geometrical methods.