On the measure of the cut locus of a Fréchet mean. (English) Zbl 1330.60023
Let \(\mu\) be a finite measure on a complete and connected Riemannian manifold \(M\). For \(p\geq 1\), the \(p\)-energy function \(F_{\mu, p}\) of \(\mu\) is defined by
\[
F_{\mu ,p}(x) = \frac{1}{p}\int_Md(x,y)^p\, d\mu (y),\quad x\in M.
\]
Assume that \(F_{\mu ,p}\) achieves a local minimum at \(x_0\). If \(p=2\) (and \(\mu\) is a probability measure), the point \(x_0\) is the Fréchet mean of \(\mu\). Assume further that, for any \(x\) in the cut locus \(C(x_0)\) of \(x_0\), there are at least two minimal geodesics from \(x_0\) to \(x\). The authors then show that \(\mu (C(x_0))=0\) (in the case \(p=1\), the further assumption \(\mu(\{x_0\})=0\) is added).
The theorem generalizes a result of T. Hotz and S. Huckemann [Ann. Inst. Stat. Math. 67, No. 1, 177–193 (2015; Zbl 1331.62269)] which treated the case of the circle \(M\) and \(p=2\). The property \(\mu (C(x_0))=0\) is an essential condition in the Central Limit Theorem for empirical Fréchet means obtained in [W. S. Kendall and H. Le, Braz. J. Probab. Stat. 25, No. 3, 323–352 (2011; Zbl 1234.60025)].
The theorem generalizes a result of T. Hotz and S. Huckemann [Ann. Inst. Stat. Math. 67, No. 1, 177–193 (2015; Zbl 1331.62269)] which treated the case of the circle \(M\) and \(p=2\). The property \(\mu (C(x_0))=0\) is an essential condition in the Central Limit Theorem for empirical Fréchet means obtained in [W. S. Kendall and H. Le, Braz. J. Probab. Stat. 25, No. 3, 323–352 (2011; Zbl 1234.60025)].
Reviewer: Wolfgang Weil (Karlsruhe)
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C22 | Geodesics in global differential geometry |
| 60F05 | Central limit and other weak theorems |
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