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On asymptotic volume of tori. (English) Zbl 0846.53043
The authors investigate the asymptotic behaviour of volume of tori in Riemannian spaces. Given a $$\mathbb{Z}^n$$-periodic metric, the main term of the volume of a ball of radius $$R$$, when $$R \to \infty$$, in such a metric, is $$cR^n$$ for some $$c > 0$$. The authors prove that $$c \geq c_E$$, where $$c_E$$ is the constant for a flat metric, and equality holds if and only if the metric is flat.
The proof is based on two lemmas. The first one generalizes a construction given by I. K. Babenko [Math. Notes 44, No. 1/2, 579-586 (1988); translation from Mat. Zametki, 44, No. 2, 177-190 (1988; Zbl 0666.52005)], and, geometrically, it is equivalent to say that for any symmetric convex body $$K$$ there exist supporting hyperplanes such that there exists an ellipsoid tangent to such hyperplanes which is inscribed in $$K$$. By this lemma the authors obtain estimates of the norm with $$K$$ as unit ball. The second lemma provides a generalized Besicovitch inequality.
They also prove that if a sequence $$d_k$$ of Riemannian metrics on a smooth manifold $$M$$ converges to a Finsler metric $$d$$ (the convergence is uniform on compact subsets of $$M\times M)$$, then the volume is not increasing, i.e. $$\text{Vol} (M,d) \leq \lim \inf \text{Vol} (M,d_k)$$ and equality implies that $$d$$ is a Riemannian metric.
The authors add some suggestions for further studies of this problem in Finsler spaces.

##### MSC:
 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
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##### References:
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