## Volume of the Minkowski sums of star-shaped sets

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Matthieu Fradelizi, Zsolt Lángi and Artem Zvavitch
**HTML**| PDF - Proc. Amer. Math. Soc. Ser. B
**9**(2022), 358-372

## Abstract:

For a compact set $A \subset \mathbb {R}^d$ and an integer $k\ge 1$, let us denote by \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots , a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of $k$ copies of $A$. A theorem of Shapley, Folkmann and Starr (1969) states that $\frac {1}{k}A[k]$ converges to the convex hull of $A$ in Hausdorff distance as $k$ tends to infinity. Bobkov, Madiman and Wang [*Concentration, functional inequalities and isoperimetry*, Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of $\frac {1}{k}A[k]$ is nondecreasing in $k$, or in other words, in terms of the volume deficit between the convex hull of $A$ and $\frac {1}{k}A[k]$, this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if $d=1$ but fails for any $d \geq 12$. In this paper we show that the conjecture is true for any star-shaped set $A \subset \mathbb {R}^d$ for $d=2$ and $d=3$ and also for arbitrary dimensions $d \ge 4$ under the condition $k \ge (d-1)(d-2)$. In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in $\mathbb {R}^d$, for any $d \geq 7$.

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## Additional Information

**Matthieu Fradelizi**- Affiliation: LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77447 Marne-la-Vallée, France
- MR Author ID: 626525
- Email: matthieu.fradelizi@u-pem.fr
**Zsolt Lángi**- Affiliation: Morphodynamics Research Group and Department of Geometry, Budapest University of Technology, Egry József utca 1, Budapest 1111, Hungary
- Email: zlangi@math.bme.hu
**Artem Zvavitch**- Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
- MR Author ID: 671170
- ORCID: 0000-0001-6347-5386
- Email: zvavitch@math.kent.edu
- Received by editor(s): October 14, 2019
- Received by editor(s) in revised form: July 12, 2021
- Published electronically: August 29, 2022
- Additional Notes: The first author was supported in part by the Agence Nationale de la Recherche, projet ASPAG - ANR-17-CE40-0017; the second author was partially supported by the National Research, Development and Innovation Office, NKFI, K-119670, the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and grants BME FIKP-VÍZ and ÚNKP-19-4 New National Excellence Program by the Ministry of Innovation and Technology; the third author was supported in part by the U.S. National Science Foundation Grant DMS-1101636 and the Bézout Labex funded by ANR, reference ANR-10-LABX-58.
- Communicated by: Deane Yang
- © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B
**9**(2022), 358-372 - MSC (2020): Primary 52A40; Secondary 52A38, 60E15
- DOI: https://doi.org/10.1090/bproc/97
- MathSciNet review: 4474695