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Formal Adventures in Convex and Conical Spaces

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Intelligent Computer Mathematics (CICM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12236))

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Abstract

Convex sets appear in various mathematical theories, and are used to define notions such as convex functions and hulls. As an abstraction from the usual definition of convex sets in vector spaces, we formalize in Coq an intrinsic axiomatization of convex sets, namely convex spaces, based on an operation taking barycenters of points. A convex space corresponds to a specific type that does not refer to a surrounding vector space. This simplifies the definitions of functions on it. We show applications including the convexity of information-theoretic functions defined over types of distributions. We also show how convex spaces are embedded in conical spaces, which are abstract real cones, and use the embedding as an effective device to ease calculations.

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Notes

  1. 1.

    The support of a probability distribution d is the set \(\{i \mid d_i > 0\}\).

  2. 2.

    This way of restricting the domain of functions in their properties rather than in the definitions is a design choice often found in Coq. It makes it possible for functions such as the logarithm to be composable without being careful about their domains and ranges, and leads to a clean separation between definitions and properties of functions in the formalization.

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Acknowledgments

We acknowledge the support of the JSPS KAKENHI Grant Number 18H03204. We also thank Shinya Katsumata for his comments.

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Authors and Affiliations

  1. National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

    Reynald Affeldt

  2. Nagoya University, Nagoya, Japan

    Jacques Garrigue & Takafumi Saikawa

Authors
  1. Reynald Affeldt
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  2. Jacques Garrigue
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  3. Takafumi Saikawa
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Corresponding author

Correspondence to Jacques Garrigue .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin, Germany

    Christoph Benzmüller

  2. National Institute of Standards and Technology, Gaithersburg, MD, USA

    Bruce Miller

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Affeldt, R., Garrigue, J., Saikawa, T. (2020). Formal Adventures in Convex and Conical Spaces. In: Benzmüller, C., Miller, B. (eds) Intelligent Computer Mathematics. CICM 2020. Lecture Notes in Computer Science(), vol 12236. Springer, Cham. https://doi.org/10.1007/978-3-030-53518-6_2

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  • DOI: https://doi.org/10.1007/978-3-030-53518-6_2

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