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Banach-Mazur Distance

The Banach-Mazur distance between two isomorphic finite-dimensional normed spaces X and Y is

d_(BM)(X,Y)=inf_(T)||T||||T^(-1)||,
(1)

where the infimum is over invertible linear transformations T:X->Y and the norms are operator norms. For origin-symmetric convex bodies K,L subset R^n, the equivalent definition is

d_(BM)(K,L)=inf{a>=1:TL subset K subset aTL, T in GL_n(R)}.
(2)

It measures distortion under linear changes of coordinates (Szarek 1990).

Let R_infty(n) be the supremum of d_(BM)(K,[-1,1]^n) over these n-dimensional convex bodies. Friedland (2026) reports the lower bound

R_infty(n)>=(cn^(2/3))/((ln(2n))^(10)),
(3)

which, combined with the upper bound of Friedland et al. (2026), would establish R_infty(n)=n^(2/3+o(1)). Here c>0 is an absolute constant and o denotes little-o notation. The new lower bound depends on a key proposition whose proof Friedland (2026) states was generated by ChatGPT at the author's request. The claimed sharp exponent remains qualified pending independent verification of that argument.

See also

Banach Space, Convex Body, Normed Space

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References

Friedland, O. "The Sharp Exponent for the Banach-Mazur Distance to the Cube." 27 Oct 2026. https://arxiv.org/abs/2610.36917.Friedland, O.; Naor, A.; and Youssef, P. "On the Banach-Mazur Radius of l_infty^n." 29 Aug 2026. https://arxiv.org/abs/2608.21112.Szarek, S. J. "Spaces with Large Distance to l_infty^n and Random Matrices." Amer. J. Math. 112, 899-942, 1990.

Cite this as:

Weisstein, Eric W. "Banach-Mazur Distance." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Banach-MazurDistance.html

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