On the generalized roundness of finite metric spaces. (English) Zbl 0842.54034

The generalized roundness introduced by P. Enflo for a metric space \((X,d)\) is the supremum \(gr (X,d)\) of all numbers \(q \geq 0\) such that \(\forall_n \geq 2\), \(\forall a_1, \dots, a_n\), \(b_1, \dots, b_n \in X\) one has: \[ \sum_{1 \leq i < j \leq n} \bigl( d(a_i, a_j)^q + d(b_i, b_j)^q \bigr) \leq \sum_{1 \leq i,j \leq n} d(a_i, b_j)^q. \] The author shows that for a finite metric space \((X,d)\), \(gr(X,d)\) can be bounded from below by a positive constant \(K(N)\) depending only upon the number of points \(N\) of \(X\): \(gr (X,d) \geq K(N)\). The best \(K(4)\) equals 1 but the best \(K(N)\) is unknown in general.


54E35 Metric spaces, metrizability
46B20 Geometry and structure of normed linear spaces
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