Local Lipschitz property for the Chebyshev center mapping over N-nets. (English) Zbl 1199.54169

Ivanshin, Pyotr N.; Sosov, Evgenii N.

Mat. Vesn. 60, No. 1, 9-22 (2008).

Let \((X,\rho)\) be a metric space and \(N\) a positive integer. \(\Sigma_N(X)\) is the set of all nonempty subsets \(Y\) of \(X\) with at most \(N\) points, and \(\Sigma_N^*(X)=\Sigma_N(X)\setminus\Sigma_{N-1}(X)\). Elements of \(\Sigma_N(X)\) (\(\Sigma_N^*(X)\)) are called \(N\)-nets (exact \(N\)-nets). For \(Y\in\Sigma_N(X)\), the point \(\operatorname{cheb}(Y)\in X\) is a Chebyshev center of \(Y\) if \(\sup\{\rho(x,\operatorname{cheb}(Y)):x\in Y\}=\inf\{\sup\{\rho(y,x):y\in Y\}:x\in X\}\). The authors investigate local Lipschitz properties of the mappings \(\operatorname{cheb}:\Sigma_N(X)\to X\), \(Y\to\operatorname{cheb}(Y)\), and \(\operatorname{cheb}| _{\Sigma_N^*(X)}\), when \(X\) is either Euclidean or Lobachevsky space, and \(\Sigma_N(X)\) is equipped with the Hausdorff metric.

Reviewer: Ljubiša Kočinac (Niš)

MSC:

54E40	Special maps on metric spaces
52C35	Arrangements of points, flats, hyperplanes (aspects of discrete geometry)

Keywords:

\(N\)-net; exact \(N\)-net; Chebyshev mapping; Hausdorff metric