Geometriae Dedicata

, Volume 47, Issue 3, pp 327–362

Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs

Authors

  • Yuri I. Lyubich
    • Dept. of MathematicsTechnion-Israel Institute of Technology
  • Leonid N. Vaserstein
    • Dept. of MathematicsPenn State University
Article

DOI: 10.1007/BF01263664

Cite this article as:
Lyubich, Y.I. & Vaserstein, L.N. Geom Dedicata (1993) 47: 327. doi:10.1007/BF01263664

Abstract

If an isometric embeddinglpmlqn with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifnN(m, q) where
$$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$
To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)⩾11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ∼ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).

Copyright information

© Kluwer Academic Publishers 1993