Abstract
If an isometric embeddingl mp →l nq with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn⩾N(m, q) where
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).
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This research was supported in part by NSF Grant DMS-9000584.
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Lyubich, Y.I., Vaserstein, L.N. Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs. Geom Dedicata 47, 327–362 (1993). https://doi.org/10.1007/BF01263664
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DOI: https://doi.org/10.1007/BF01263664