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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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