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Obtuse constants of Alexandrov spaces. (English) Zbl 1443.53019
In [K. Grove and P. Petersen, Ann. Math. (2) 128, No. 1, 195–206 (1988; Zbl 0655.53032)], it is shown that there exist constants $$\delta$$ and $$\epsilon$$, depending on the dimension, curvature, volume and diameter of a Riemannian manifold, so that, if points $$p$$ and $$q$$ are $$\delta$$-close on the manifold, it is possible to draw a geodesic from one of the points which makes an obtuse angle of at least $$\frac{\pi}{2} + \epsilon$$ with any shortest geodesic between $$p$$ and $$q$$. This same property holds true more generally in Alexandrov spaces.
The present paper asks whether, for a close pair $$p, q$$, there is a third point $$x$$ lying far from $$p$$ and $$q$$ with a shortest path from $$x$$ to one of $$p$$ or $$q$$ that creates an obtuse angle with any shortest path between $$p$$ and $$q$$. We may take ‘far’ to mean ‘lying at least half the radius of $$M$$ from the point where the angle is subtended’.
This is measured by a new geometric invariant. The obtuse constant of a pair gives a lower bound for the size of the obtuse angle which can be created by the best choice of $$x$$. The obtuse constant of $$M$$ itself is defined by taking the limit inferior as the distance between $$p$$ and $$q$$ approaches zero.
The obtuse constant is positive and is bounded below in terms of the dimension, curvature, volume and diameter, generalising the result of Grove and Petersen [loc. cit.]. The authors investigate its relationship to the volume, volume growth rate and asymptotic cone and also obtain rigidity results.
##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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