Generalized roundness and negative type
A metric space \((X,d)\) has roundness \(q\) if, for every pair of pairs of points \(\{a_1,a_2\}\) and \(\{b_1,b_2\}\) in \(X\) we have \[ d(a_1,a_2)^q + d(b_1,b_2)^q \leq \sum_{1\leq 1,j \leq 2} d(a_i,b_j)^q. \] Generalized roundness is defined similarly by means of an inequality for each pair of \(n\)-tuples of points. The space has negative type \(q\) if for every \(n \in \mathbb N\) and each \(n\)-tuple \(\{a_1, \ldots , a_n\}\) in \(X\) and real numbers \(\{\xi_1, \ldots , \xi_n\}\) with \(\sum \xi_i = 0\) we have \(\sum_{1\leq 1,j \leq n} d(a_i,a_j)^q \xi_i\xi_j \leq 0.\) The main result is that \((X,d)\) has \(q\)-negative type if and only if it has generalized roundness \(q\). A corollary to this and a known result [\textit{J. H. Wells} and \textit{L. R. Williamson}, ``Embeddings and extensions in analysis'' (1975; Zbl 0324.46034)] is that if \(p > 2\) and \(q > 0\) then \(L_p(\Omega, \Sigma, \mu)\) (of dimension \(\geq 3\)) does not have generalized roundness \(q\). Another corollary is that if \(0 < p \leq 2\) then \(L_p\) has generalized roundness \(q\) if and only if \(q \in [0,p]\).
scientific article; zbMATH DE number 1046966