Reverse Brunn-Minkowski constant
Description of constant
For subsets $K,L\subset\mathbb{R}^n$, their Minkowski sum is
\[K+L\ :=\ \{x+y:\ x\in K,\ y\in L\}.\]In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant. [Bas1995-no-general-reverse]
Milman’s reverse Brunn-Minkowski theorem says that for centrally symmetric convex bodies, after putting the bodies in a suitable relative position, one does get a dimension-free reverse inequality. [Bas1995-milman-statement]
More precisely, define $C_{\mathrm{RBM}}$ as the infimum of constants $C>0$ such that for every $n\ge 1$ and every pair of centrally symmetric convex bodies $B_1,B_2\subset\mathbb{R}^n$, there exists a linear map $u$ with $\lvert\det u\rvert=1$ satisfying
\[\lvert u(B_1)+B_2\rvert^{1/n}\ \le\ C\bigl(\lvert B_1\rvert^{1/n}+\lvert B_2\rvert^{1/n}\bigr).\]We define
\(C_{70}\ :=\ C_{\mathrm{RBM}},\) the reverse Brunn-Minkowski constant.
Milman’s theorem implies $C_{\mathrm{RBM}}<\infty$. [Bas1995-milman-statement]
A trivial lower bound is $C_{\mathrm{RBM}}\ge 1$.
Thus the best established range currently is
\[1\ \le\ C_{70}\ <\ \infty.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $<\infty$ | [Mil1986], [Bas1995] | Existence of a dimension-free constant in Milman’s theorem. [Bas1995-milman-statement] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $1$ | Trivial scaling lower bound. |
Additional comments and links
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Original source. The reverse inequality was introduced by Milman in 1986. [Mil1986]
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Proof history. Bastero-Bernues-Pena explicitly record that Pisier (1989) and Milman (1988) gave alternative proofs of Milman’s theorem. [Bas1995-proof-history]
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Published extension. Bastero-Bernues-Pena prove a broader affine-invariant reverse inequality for bodies with $p(A_1),p(A_2)\ge p$ (with constant depending only on $p$). [Bas1995-extension-theorem1]
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Wikipedia page on Milman’s reverse Brunn-Minkowski inequality
References
- [Bas1995] Bastero, J.; Bernues, J.; Pena, A. An extension of Milman’s reverse Brunn-Minkowski inequality. Geometric and Functional Analysis 5 (1995), no. 3, 572-581. DOI: https://doi.org/10.1007/BF01895832. Publisher page: https://link.springer.com/article/10.1007/BF01895832. arXiv PDF: https://arxiv.org/pdf/math/9501210.pdf. Google Scholar
- [Bas1995-no-general-reverse] loc: arXiv PDF p.1, Introduction quote: “It is easy to see that one cannot expect the reverse inequality to hold at all, even if it is perturbed by a fixed constant and we restrict ourselves to balls…”
- [Bas1995-milman-statement] loc: arXiv PDF p.1, Introduction (quoted Milman theorem statement) quote: “There exists a constant $C > 0$ such that for all $n \in \mathbb{N}$ and any balls $B_1, B_2 \subset \mathbb{R}^n$ we can find a linear transformation $u: \mathbb{R}^n \to \mathbb{R}^n$ with $|\det(u)| = 1$ and $|u(B_1) + B_2|^{1/n} \le C(|B_1|^{1/n} + |B_2|^{1/n})$.”
- [Bas1995-proof-history] loc: arXiv PDF p.1, Introduction quote: “Pisier in [Pi 2] gave a new proof by using interpolation and entropy estimates. Milman in [Mil 2] gave another proof by using the ‘convex surgery’ and achieving also some entropy estimates.”
- [Bas1995-extension-theorem1] loc: arXiv PDF p.1, Theorem 1 quote: “Let $0 < p \le 1$. There exists $C = C(p) \ge 1$ such that for all $n \in \mathbb{N}$ and all $A_1, A_2 \subset \mathbb{R}^n$ bodies such that $p(A_1), p(A_2) \ge p$, there exists an affine transformation $T$ … such that $|T(A_1) + A_2|^{1/n} \le C(|A_1|^{1/n} + |A_2|^{1/n})$.”
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[Mil1986] Milman, V. D. Inegalite de Brunn-Minkowsky inverse et applications a la theorie locale des espaces normes. C. R. Acad. Sci. Paris, Serie I 302 (1986), 25-28. MR: 0827101. Google Scholar
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[Mil1988] Milman, V. D. Isomorphic symmetrization and geometric inequalities. In Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Mathematics 1317 (1988), 107-131. DOI: https://doi.org/10.1007/BFb0081738. Google Scholar
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[Pis1989] Pisier, Gilles. The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press (1989). DOI: https://doi.org/10.1017/CBO9780511662454. Google Scholar
- [Pis1989b] Pisier, Gilles. A new approach to several results of V. Milman. Journal fur die reine und angewandte Mathematik 393 (1989), 115-131. DOI: https://doi.org/10.1515/crll.1989.393.115. Google Scholar
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.