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68 3.9 The pseudodistance and Toponogov's theorem Chapter 3 We use the notation of section 3.5. By Proposition 3.5.1, the fixed point set F of the map -1 1 in X is just i a X (). It follows that if x, Y T X X , x X (), then -1 1 x, Y T X = x, Y T X . / For a R, let a be the dilation a x, Y T X = x, aY T X . Since for t = 0, t = 1/t 1 t , we deduce from the above that if x X (), for any t R, / -1 t x, Y T X = x, Y T X . Take x, Y TX (3.9.1) (3.9.2) (3.9.3) X , x X (). In the sequel, we will assume that / Y T X = 1. (3.9.4) Set x , Y TX = x, Y T X . (3.9.5) From the above, it follows that for any t R, t/2 x, Y T X = -t/2 x , Y T X . Let s [0, 1] x s X be the geodesic connecting x to x . Set Y 0 T X = x 0 , Then x , Y 1 T X = 1 x, Y 0 T X . Moreover, d (x) = Y 0 T X = Y 1 T X . By equation (3.4.32), d (x) = 1 Y 1 T X - Y 0 T X . d (x) (3.9.10) (3.9.9) (3.9.8) Y 1 T X = x 1 . (3.9.7) (3.9.6) Assume that x = (1, f ) , f p (). By (3.4.4), (3.4.5), and (3.9.10), for |f | 1, Y 1 T X - Y 0 T X C d (x) C (|a| + C |f |) . Moreover, by (3.4.6) and (3.9.10), for |f | 1, Y 1 T X - Y 0 T X C |f | . (3.9.12) (3.9.11) Theorem 3.9.1. Given > 0, there exists C , > 0 such that if x X is such that d (x, X ()) , if Y T X T x X, Y T X = 1, for t 0, then t x, Y T X , -t x, Y T X C , . (3.9.13)