×

Means in metric spaces and the center of mass. (English) Zbl 1225.26062

Let \(\left( X,d\right) \) be a metric space. The length \(L(\gamma )\) of a curve \(\gamma :I\rightarrow X\) is defined as the supremum of \( \sum_{i=1}^{n}d\left( \gamma \left( t_{i-1}\right) ,\gamma \left( t_{i}\right) \right) \) where \(t_{0}\leq t_{1}\leq \cdots \leq t_{n}\in I.\) A curve \(\gamma :I\rightarrow X\) is called a geodesic if \(d\left( \gamma \left( s\right) ,\gamma \left( t\right) \right) =d\left( \gamma \left( s\right) ,\gamma \left( r\right) \right) +d\left( \gamma \left( r\right) ,\gamma \left( t\right) \right) \) for all \(s<r<t\in I.\) The metric space \(X\) is called a geodesic metric space if any two points \(x,y\) can be connected with a geodesic \(\gamma \) such that \(L(\gamma )=d(x,y).\) A geodesic metric space is called uniquely geodesic if any two points \(x,y\) can be connected by a unique geodesic \(\gamma _{x,y}\). It follows that we have unique metric midpoints, the midpoints of these geodesics, denoted as \(x\sharp y=\gamma _{x,y}(1/2)\). A geodesic metric space is \(k\)-convex if for any three points \( x,y,z,\) any geodesic \(\gamma :[0,1]\rightarrow X\) between \(x\) and \(y\) and for all \( t\in [0,1]\) we have
\[ d(z,\gamma (t))^{2}\leq (1-t)d(z,x)^{2}+td(z,y)^{2}-\frac{k}{2} t(1-t)d(x,y)^{2}. \]
A \(k\)-convex space is uniquely geodesic. The author gives an extension of midpoint maps as means between two points to several variables. Let \((X,d)\) be a complete \(k\)-convex geodesic metric space. Let \(Q_{1}^{0},\dots,Q_{n}^{0}\) be points in \(X\) and \(\pi =(\pi _{0},\pi _{1},\dots)\) be an infinite sequence of permutations of the letters \(\{1,\dots,n\}\). Let
\[ Q_{i}^{l+1}=\begin{cases} Q_{\pi _{i}(i)}^{l}\sharp Q_{\pi _{i}(i+1)}^{l}\;\text{if }1\leq i<n \\ Q_{\pi _{i}(n)}^{l}\sharp Q_{\pi _{i}(1)}^{l}\qquad \text{else.} \end{cases} \]
Then the sequences \(Q_{i}^{l}\) converge to a common limit point which is the centre of mass of the points \(Q_{1}^{0},\dots,Q_{n}^{0}\) defined by \(\underset{ x\in X}{\arg \min }\sum_{i=1}^{n}d(x,Q_{i}^{0})^{2},\) where \(\underset{x\in X }{\arg \min }C(x)\) denotes the unique point that minimizes the function \( C(x).\) The case of the symmetric space on the convex cone of positive definite matrices related to the geometric mean and the special orthogonal group are also studied as examples of \(k\)-convex metric spaces.

MSC:

26E60 Means
53C20 Global Riemannian geometry, including pinching
54E35 Metric spaces, metrizability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ando, T.; Li, C.-K.; Mathias, R., Geometric means, Linear Algebra Appl., 385, 305-334 (2004) · Zbl 1063.47013
[2] Burago, D.; Burago, Y.; Ivanov, S., A Course in Metric Geometry (2001), Amer. Math. Soc. · Zbl 0981.51016
[3] Ballmann, W., Lectures on Spaces of Nonpositive Curvature (1995), Birkhäuser-Verlag: Birkhäuser-Verlag Basel · Zbl 0834.53003
[4] Bhatia, R., Matrix Analysis (1996), Springer-Verlag: Springer-Verlag New York
[5] Bhatia, R.; Holbrook, J., Riemannian geometry and matrix geometric means, Linear Algebra Appl., 413, 594-618 (2006) · Zbl 1088.15022
[6] Bini, D. A.; Meini, B.; Poloni, F., An effective matrix geometric mean satisfying the Ando-Li-Mathias properties, Math. Comp., 79, 269, 437-452 (2010) · Zbl 1194.65065
[7] Bridson, M. R.; Haefliger, A., Metric Spaces of Non-Positive Curvature (1999), Springer-Verlag · Zbl 0988.53001
[8] Hiai, F.; Kosaki, H., Means of Hilbert Space Operators, Lecture Notes in Math., vol. 1820 (2003), Springer · Zbl 1048.47001
[9] Jost, J., Nonpositive Curvature: Geometric and Analytic Aspects, Lectures Math. ETH Zurich (1997), Birkhäuser-Verlag: Birkhäuser-Verlag Basel · Zbl 0896.53002
[10] Jung, C.; Lee, H.; Yamazaki, T., On a new construction of geometric mean of \(n\)-operators, Linear Algebra Appl., 431, 1477-1488 (2009) · Zbl 1172.47017
[11] Karcher, H., Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30, 509-541 (1977) · Zbl 0354.57005
[12] Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 205-224 (1980) · Zbl 0412.47013
[13] Lawson, J.; Lim, Y., A general framework for extending means to higher orders, Colloq. Math., 113, 191-221 (2008) · Zbl 1160.47016
[14] J.H. Manton, A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups, in: Eighth International Conference on Control, Automation, Robotics and Vision, Kunming, China, December 2004.; J.H. Manton, A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups, in: Eighth International Conference on Control, Automation, Robotics and Vision, Kunming, China, December 2004.
[15] Moakher, M., A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26, 735-747 (2005) · Zbl 1079.47021
[16] Moakher, M., Means and averaging in the group of rotations, SIAM J. Matrix Anal. Appl., 24, 1, 1-16 (2002) · Zbl 1028.47014
[17] Ohta, S.-I., Convexities of metric spaces, Geom. Dedicata, 125, 225-250 (2007) · Zbl 1140.52001
[18] M. Pálfia, A multi-variable extension of two-variable matrix means, SIAM J. Matrix Anal. Appl. (2011), in press.; M. Pálfia, A multi-variable extension of two-variable matrix means, SIAM J. Matrix Anal. Appl. (2011), in press.
[19] Papadopoulos, A., Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lect. Math. Theor. Phys., vol. 6 (2005), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1115.53002
[20] Simmons, G. F., Introduction to Topology and Modern Analysis (1966), McGraw-Hill Book Company Inc.
[21] Sturm, K. T., Probability measures on metric spaces of nonpositive curvature, (Auscher, Pascal; etal., Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs. Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs, April 16-July 13, 2002, Paris, France. Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs. Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Lecture Notes from a Quarter Program on Heat Kernels, Random Walks, and Analysis on Manifolds and Graphs, April 16-July 13, 2002, Paris, France, Contemp. Math., vol. 338 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 357-390 · Zbl 1040.60002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.