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On the volume of metric balls

Author: Christopher B. Croke
Journal: Proc. Amer. Math. Soc. 88 (1983), 660-664
MSC: Primary 53C20; Secondary 53C22
MathSciNet review: 702295
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Abstract: In this paper we consider metrics of the form $ d{s^2} = d{r^2} + {f^2}(r,\theta )d{\theta ^2}$ on a ball of dimension $ n \geqslant 3$. We show that if the diameters (geodesics through the origin) minimize length then the volume of the ball is larger than the volume of the hemisphere of the corresponding round sphere. This relates to a conjecture first considered by Marcel Berger. We also give examples in all dimensions of radially symmetric metrics on balls of radius $ \pi /2$ having arbitrarily small volume and yet having no pair of points conjugate along a diameter.

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Article copyright: © Copyright 1983 American Mathematical Society