Issue |
ESAIM: M2AN
Volume 58, Number 1, January-February 2024
|
|
---|---|---|
Page(s) | 393 - 420 | |
DOI | https://doi.org/10.1051/m2an/2023092 | |
Published online | 28 February 2024 |
On the numerical approximation of Blaschke–Santaló diagrams using Centroidal Voronoi Tessellations
1
Centre de Mathématiques Appliquées, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
2
Dipartimento di Matematica, Università di Pisa Largo B. Pontecorvo 5, 56127 Pisa, Italy
3
Laboratoire Jean Kuntzmann (LJK), Université Joseph Fourier Tour IRMA, BP 53, 51 rue des Mathématiques, 38041 Grenoble Cedex 9, France
* Corresponding author: edouard.oudet@imag.fr
Received:
21
February
2023
Accepted:
7
November
2023
Blaschke–Santaló diagrams are images of maps defined on a set of parameters, taking values into an Euclidean space. Typically, the dimension of the source space is high, possibly infinite, while the target space is two or three dimensional. These diagrams help characterize geometrically various inequalities and are of particular interest in the field of shape optimization. We propose a numerical method, based on Centroidal Voronoi Tessellations, which produces sample points in the parameter space that have uniformly distributed images in the Blaschle–Santaló diagram, therefore providing an accurate description of the latter. Compared with the classical Monte Carlo methods, which simply use a large number of images corresponding to random parameters, the method proposed is computationally efficient and precise. Simulations for two and three dimensional diagrams are presented involving examples in algebra and shape optimization.
Mathematics Subject Classification: 49Q10 / 49M20 / 65C05 / 65C20 / 52B35
Key words: Blaschke–Santaló diagrams / Voronoi tessellations / Monte Carlo methods / optimal transport / Lloyd’s algorithm
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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