## 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.

### MSC:

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

### Keywords:

weighted simplex; generalized roundness
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