Reverse Brunn-Minkowski constant ($C_{70}$)

• Original source. The reverse inequality was introduced by Milman in 1986. [Mil1986]

• Proof history. Bastero-Bernues-Pena explicitly record that Pisier (1989) and Milman (1988) gave alternative proofs of Milman's theorem. [Bas1995-proof-history]

• 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]

• Wikipedia page on Milman's reverse Brunn-Minkowski inequality

References (from the repository page)

• [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 $|T(A_1) + A_2|^{1/n} \le C(|A_1|^{1/n} + |A_2|^{1/n})$."

• [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

• [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

• [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

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound $<\infty$. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds