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Total diameter and area of closed submanifolds

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Abstract

The total diameter of a closed planar curve \(C\subset \mathbf {R}^2\) is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of \(C\). Furthermore, when \(C\) is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when \(C\) is a circle. We also generalize these results to \(m\) dimensional submanifolds of \(\mathbf {R}^n\), where the “area” will be defined in terms of the mod \(2\) winding numbers of the submanifold about the \(n-m-1\) dimensional affine subspaces of \(\mathbf {R}^n\).

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Acknowledgments

We are grateful to Jaigyoung Choe who first suggested to us that inequality (3) should hold for planar curves, and thus provided the initial stimulus for this work. We also thank Igor Belegradek for locating the reference [21]. Finally, thanks to the anonymous referee for suggesting improvements to an earlier draft of this work.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Mohammad Ghomi

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Ralph Howard

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  1. Mohammad Ghomi
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  2. Ralph Howard
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Correspondence to Mohammad Ghomi.

Additional information

M. Ghomi was supported in part by NSF Grants DMS-1308777, DMS-0806305, and Simons Collaboration Grant 279374.

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Ghomi, M., Howard, R. Total diameter and area of closed submanifolds. Math. Ann. 363, 985–999 (2015). https://doi.org/10.1007/s00208-015-1173-4

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