Isosystolic inequalities for optical hypersurfaces

Abstract

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encloses a volume $V$, then it carries a periodic characteristic whose action is at most $\sqrt{V/3}$. This result is deduced from an interesting dual version of Minkowski's lattice-point theorem: if the origin is the unique integer point in the interior of a planar convex body, the area of its dual body is at least 3/2.

36 pages, 2 figures

References in corpus (3)

• Diastolic and isoperimetric inequalities on surfaces
• Measurements of Riemannian two-disks and two-spheres
• A note on Mahler's conjecture

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• Measurements of Riemannian two-disks and two-spheres
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• Discrete surfaces with length and area and minimal fillings of the circle
• Minkowski's successive minima in convex and discrete geometry
• Improvement and generalisation of Papasoglu's lemma