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Convex Hulls of Curves: Volumes and Signatures

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Geometric Science of Information (GSI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14071))

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Abstract

Taking the convex hull of a curve is a natural construction in computational geometry. On the other hand, path signatures, central in stochastic analysis, capture geometric properties of curves, although their exact interpretation for levels larger than two is not well understood. In this paper, we study the use of path signatures to compute the volume of the convex hull of a curve. We present sufficient conditions for a curve so that the volume of its convex hull can be computed by such formulae. The canonical example is the classical moment curve, and our class of curves, which we call cyclic, includes other known classes such as d-order curves and curves with totally positive torsion. We also conjecture a necessary and sufficient condition on curves for the signature volume formula to hold. Finally, we give a concrete geometric interpretation of the volume formula in terms of lengths and signed areas.

Supported by NCCR-Synapsy Phase-3 SNSF grant number 51NF40-185897 and Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA).

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Acknowledgements

We are grateful to Bernd Sturmfels, Anna-Laura Sattelberger, and Antonio Lerario for helpful discussions.

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Authors and Affiliations

  1. Technical University of Berlin, Berlin, Germany

    Carlos Améndola

  2. University of Oxford, Oxford, UK

    Darrick Lee

  3. MPI-MiS Leipzig, Leipzig, Germany

    Chiara Meroni

Authors
  1. Carlos Améndola
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  2. Darrick Lee
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  3. Chiara Meroni
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Corresponding author

Correspondence to Carlos Améndola .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories Inc., Tokyo, Japan

    Frank Nielsen

  2. THALES Land and Air Systems, Meudon, France

    Frédéric Barbaresco

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Améndola, C., Lee, D., Meroni, C. (2023). Convex Hulls of Curves: Volumes and Signatures. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_45

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  • DOI: https://doi.org/10.1007/978-3-031-38271-0_45

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-38270-3

  • Online ISBN: 978-3-031-38271-0

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