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Bourgain’s Slicing Problem

Determine whether there exists a universal constant c > 0 such that for every integer n ≥ 2 and every convex body K ⊂ ℝ^n with Vol_n(K) = 1, there exists a hyperplane H ⊂ ℝ^n with Vol_{n−1}(K ∩ H) > c.

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Background

Bourgain’s slicing problem asks for a uniform lower bound on the maximal volume of hyperplane sections of a unit-volume convex body. It is equivalent to a dimension-free upper bound on the isotropic constant of convex bodies and is tightly connected to central results in asymptotic convex geometry.

The best known general bound replaces the universal constant by c/√(log n). Establishing a dimension-free bound would resolve several related problems and sharpen volumetric and isoperimetric inequalities for convex bodies.

References

Let n ≥ 2 and suppose that K ⊂ ℝn is a convex body of volume one. Does there exist a hyperplane H ⊂ ℝn such that Vol_{n−1}(K ∩ H) > c for a universal constant c > 0? This question is still not completely answered, and in the last four decades it emerged as an “engine” for the development of the research direction discussed in these lectures.

— Isoperimetric inequalities in high-dimensional convex sets (2406.01324 - Klartag et al., 3 Jun 2024) in Question, Section 9 (Bourgain's slicing problem)