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Convex functions and barycenter on CAT(1)-spaces of small radii. (English) Zbl 1351.53057

It is well known that the distance function of \(\mathrm{CAT}(0)\)-spaces is convex. While this is in general not true for arbitrary \(\mathrm{CAT}(\kappa)\)-spaces with \(\kappa>0\), the paper under review shows convexity for a certain function, which is explicitly defined in terms of the distance function. That function was first considered by W. S. Kendall [J. Lond. Math. Soc., II. Ser. 43, No. 3, 567–576 (1991; Zbl 0688.58001)], who had proved its convexity for the standard sphere.
The main application of this result is to prove the existence and uniqueness of a barycenter for a probability measure \(\mu\) on a \(\mathrm{CAT}(\kappa)\)-space \(X\), assuming \(\mu\) is concentrated on a ball of radius \(\frac{\pi}{2\sqrt{\kappa}}\). Here a barycenter of \(\mu\) means a global minimum of the functon \(x\to\int_Xd^2(.,x)d\mu\).
Other applications include theorems of Banach-Saks-Kakutani type for \(\mathrm{CAT}(\kappa)\)-spaces, generalising results proved by J. Jost [Calc. Var. Partial Differ. Equ. 2, No. 2, 173–204 (1994; Zbl 0798.58021)] for \(\mathrm{CAT}(0)\)-spaces: Any sequence of points \((p_n)\) with radius bounded by \(\frac{\pi}{2\sqrt{\kappa}}\) has a subsequence, for which the sequence of barycenters of the measures \(\frac{1}{n}\sum_{i=1}^n\delta_{p_i}\) converges to a point in \(X\).

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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