The Concentration Phenomenon and Linear Structure of Finite-Dimensional Normed Spaces · 1986-ana-197

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No citable source was found that proves or disproves the universal $Cn$ estimate for all finite-dimensional real normed spaces.

Vitali D. Milman posed the inequality as Problem 6 in The Concentration Phenomenon and Linear Structure of Finite-Dimensional Normed Spaces, Proceedings of the International Congress of Mathematicians, Berkeley 1986, Vols. 1–2, American Mathematical Society, 1987, pp. 961–975 (direct PDF).

The strongest general estimate obtained from the standard $MM^*$ theory is

Indeed, if $K=u^{-1}(B_X)$, Haar averaging over orthonormal bases gives $\mathbb E_e r_e(u)=\sqrt n\,M(K)$ and $\mathbb E_\eta r_\eta((u^{-1})^*)=\sqrt n\,M(K^\circ)$. The $MM^*$ estimate supplies a linear position with $M(K)M(K^\circ)\le C\log(n+1)$, after which the two bases may be selected independently. This logarithmic $MM^*$ estimate is presented in Gilles Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics 94, Cambridge University Press, 1989, DOI 10.1017/CBO9780511662454; it combines the $\ell$-position machinery of D. R. Lewis and T. Figiel–N. Tomczak-Jaegermann with Pisier's estimate for the Rademacher projection. It leaves exactly a logarithmic gap from the requested bound. The argument does yield the desired order $Cn$ for families whose $MM^*$ product is uniformly bounded, but not for arbitrary $X$.

Stanislaw J. Szarek explicitly revisited the Rademacher-product question in Problem 3.3 of Convexity, Complexity, and High Dimensions, Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. II, EMS Press, 2007, pp. 1599–1621, DOI 10.4171/022-2/76. He states that the corresponding full-space property remains open and identifies Milman's Problem 6 as a slightly weaker formulation. This rules out treating the classical $MM^*$ estimate or proportional-subspace results as a resolution of the exact problem.

A weaker consequence also remains incomplete. Stanislaw J. Szarek and Nicole Tomczak-Jaegermann, On the Nontrivial Projection Problem, Advances in Mathematics 221 (2009), no. 2, 331–342, DOI 10.1016/j.aim.2008.12.004, arXiv:0805.3792, prove an "up to a logarithmic factor" result: every $n$-dimensional space has a complemented subspace of one of the types $\ell_1^k,\ell_2^k,\ell_\infty^k$, with $k\ge c\exp(\tfrac12\sqrt{\log n})$, but with projection/isomorphism bound $C(1+\log k)^2$, not a universal constant. Timothy Gowers and Katarzyna Wyczesany, A Counterexample to a Strengthening of a Question of V. D. Milman, Annales Henri Lebesgue 6 (2023), 427–448, DOI 10.5802/ahl.168, still describe the bounded nontrivial-projection question as open; their counterexample concerns a stronger requirement involving the fixed Euclidean structure and does not disprove the Rademacher-product inequality.