Isometric embeddings of finite-dimensional $\ell_p$-spaces over the quaternions

Abstract

Abstract. The nonexistence of isometric embeddings m q → n p with p = q is proved. The only exception is q = 2, p ∈ 2N, in which case an isometric embedding exists if n is sufficiently large, n ≥ N (m, p). Some lower bounds for N (m, p) are obtained by using the equivalence between the isometric embeddings in question and the cubature formulas for polynomial functions on projective spaces. Even though only the quaternion case is new, the exposition treats the real, complex, and quaternion cases simultaneously.

Lower bounds for projective designs, cubature formulas and related isometric embeddings

“…DEFINITION 1.1. [14] Let p be an integer even, p ≥ 2. A function φ : K m → C belongs to the class ΦK(m, p) if a) φ is a homogeneous polynomial of degree p on the real space R δm ≡ (K m )R and b) φ is U(K)-invariant in the sense that φ(xα) = φ(x), x ∈ K m , |α| = 1, or equivalently,…”

Upper bound for isometric embeddings ℓ₂^{𝑚}→ℓ_{𝑝}ⁿ

“…Let K [3] for K = R and [4] for any K. Conversely, under these conditions for m and p, there exists an n such that m 2;K can be isometrically embedded into n p;K ; see [6] (and also [5,7]) for K = R, [2] for K = C, and [4] for K = H, C and R simultaneously. The proofs of existence in these papers also yield some upper bounds for the minimal n = N K (m, p).…”

“…The proofs of existence in these papers also yield some upper bounds for the minimal n = N K (m, p). According to [4], these bounds can be joined in the inequality…”

“…Note that in all cases dim Φ K (m, p) can be explicitly expressed through binomial coefficients. (All the formulas are brought together in [4,Theorem 2]. )…”

Polynomial functions on the classical projective spaces

“…An application (cf. [5] and the references therein). We consider the problem of existence of isometric embeddings m 2 → n p or, equivalently, of existence of m-dimensional Euclidean subspaces in n p over K. The latter is K n endowed with the norm…”