Wednesday, 10 June 2009

mg.metric geometry - Stability of midpoints in CAT(0) spaces

No, even if X=mathbbR2.



Let A1 be (the convex hull of) 4 points with coordinates (pm1,pm1). Then m(A1)=(0,0), as the 4 points are on the circle S1 of radius sqrt2 centered at (0,0). Shift S1 a small distance varepsilon in the horizontal direction, denote the resulting circle by S2. For each vertex of A1, mark its nearest point on S2. The marked points are vertices of a convex quadrangle A2 inscribed in S2 and containing its center (varepsilon,0). Hence m(A2)=(varepsilon,0) but the Hausdorff distance between A1 and A2 is approxvarepsilon/sqrt2.

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