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Brunn-Minkowski Inequality

The Brunn-Minkowski inequality states that the nth root of the content of the set sum of two sets in n-dimensional Euclidean space is greater than or equal to the sum of the nth roots of the contents of the individual sets.

Liu et al. (2026) reported the logarithmic Brunn-Minkowski inequality for convex bodies whose centroid is the origin, without requiring origin symmetry, in every dimension n>=2. They state that all proofs were generated by ChatGPT; these proofs have not been independently verified here.

See also

Logarithmic Brunn-Minkowski Inequality, Tomography

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References

Cover, T. M. "The Entropy Power Inequality and the Brunn-Minkowski Inequality" §5.10 in Open Problems in Communications and Computation. (Ed. T. M. Cover and B. Gopinath). New York: Springer-Verlag, p. 172, 1987.Liu, Y.; Sun, Q.; and Xiong, G. "The Centered Logarithmic Brunn-Minkowski Inequality." 7 Oct 2026. https://arxiv.org/abs/2610.10152.Schneider, R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge, England: Cambridge University Press, 1993.

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Brunn-Minkowski Inequality

Cite this as:

Weisstein, Eric W. "Brunn-Minkowski Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Brunn-MinkowskiInequality.html

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