Brunn–Minkowski inequality for general even log-concave measures in arbitrary dimension

Establish the Brunn–Minkowski inequality for general even log-concave measures in every dimension, extending the known two-dimensional result and the currently weaker higher-dimensional estimates.

Background

The paper places its Gaussian-mixture Brunn–Minkowski inequality in the broader context of Brunn–Minkowski inequalities for even log-concave measures. It notes that the two-dimensional case has been established, while the general-dimensional problem remains unresolved.

This open problem concerns obtaining the full Brunn–Minkowski inequality, rather than a version with a weaker exponent, for arbitrary even log-concave measures in dimensions greater than two.

References

The Brunn--Minkowski inequality for general even log-concave measures is one of the central problems in contemporary convex geometry. The two-dimensional case has already been established by Livshyts--Marsiglietti--Nayar--Zvavitch , as a consequence of the two-dimensional log-Brunn--Minkowski conjecture for the Lebesgue proved by Böröczky--Lutwak--Yang--Zhang and its extension to general even log-concave measures by Saroglou . However, the general-dimensional case remains open, and the current best result, allowing a weaker exponent, is due to Yang (following prior contributions in ).

— Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality  (2610.13596 - Milman et al., 9 Oct 2026) in Section 1, subsection “Gaussian mixture inequalities”