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 $C_{\mathrm{RBM}}\ge 1$.
Including the unrefereed finite certificate below, the recorded range is
\[1.20194336\ \le\ C_{70}\ <\ \infty.\]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. | |
| $1.201927$ | [Pat2026] | Cube $B_1=[0,a]^n$ and ball $B_2=rB$ with $n=300000$, $a/r=0.013$. For $\lvert\det u\rvert=1$, $u(B_1)$ is a parallelotope; its intrinsic volumes are sums of $j$-face volumes, and Szász’s inequality on its Gram matrix with AM–GM gives $V_j(u(B_1))\ge\binom{n}{j}a^j$, so by Steiner’s formula $\lvert u(B_1)+rB\rvert\ge\sum_j\binom{n}{j}a^j\kappa_{n-j}r^{n-j}$ for every $u$. Hence $C_{70}\ge(\sum_j\binom{n}{j}a^j\kappa_{n-j}r^{n-j})^{1/n}/(a+\kappa_n^{1/n}r)$; evaluated in interval arithmetic. ($n=2$ gives $\sqrt{1/2+1/\sqrt\pi}\approx1.0316$; the value increases with $n$.) |
| $1.20194336$* | [Pin2026] | Optimization of Patankar’s cube-and-ball construction. Retain the Steiner term with $n=10^{12}$, $k=367879441172$, $t=k/n$ and $a/\kappa_n^{1/n}=(n-k)/k=632120558828/367879441172$; its normalized root is enclosed near $1.20194336844315056699>1.20194336$. Arb and exact-rational logarithm/Stirling checkers, both written by the same AI agent, pass. The paper proves the single-term limit $\exp(1/(2e))>1.20194336847$ as even $n\to\infty$ and $k/n\to1/e$; this analytic limit is separate from the finite certificate. Unrefereed, not formally verified, without independent human review or an optimality claim. |
Additional comments and links
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The finite certificate gives $C_{70}>1.20194336$; the prose limiting argument gives the stronger $C_{70}\ge\exp(1/(2e))>1.20194336847$. The asterisk records the self-archived, unrefereed verification status. The construction and positioning reduction are due to Patankar; this note optimizes a single term.
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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
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[Pat2026] Patankar, V. Rigorous lower bounds for constants 16a, 20a, 20c and 70a, and an upper bound for 23b, with verification scripts. GitHub repository (2026): https://github.com/vidigoat/optimization-records/tree/main/tao.
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[Pat2026] Patankar, V. Rigorous lower bounds for constants 16a, 20a, 20c and 70a, and an upper bound for 23b, with verification scripts, 14 October 2026. Pinned code and proofs; proposed in PR #226; independent replay by Carlos Toledo, 16 October 2026.
- [Pin2026] Pinho, Ricardo. Optimizing a cube–ball lower bound for the reverse Brunn–Minkowski constant, 23 October 2026, unrefereed preprint. Paper, certificate and both checkers. AI-assisted analysis and proof drafting under the author’s direction; author review at summary level. Not peer reviewed or formally verified; exhaustive priority is not asserted.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro. The bound from [Pat2026] was found with search and proof assistance from Claude (Anthropic); its value is recomputed by the scripts linked in that reference.