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Thin-Shell Conjecture

Show that the thin-shell constant S_n, defined by S_n = sup_X (1/n) Var(|X|^2) where the supremum is over isotropic log-concave random vectors X in ℝ^n, is bounded by a universal constant independent of n.

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Background

The thin-shell conjecture posits that the Euclidean norm of isotropic log-concave random vectors concentrates sharply around √n, with variance O(n). It is known to imply quantitative central limit theorems and is strictly weaker than KLS.

Bounds derived via stochastic localization relate S_n to spectral-gap bounds, and historically led to progress on KLS; the optimal order remains conjectural.

References

The constant S_n is called the thin-shell constant. The thin-shell conjecture asserts that the sequence (S_n) is bounded.

— Isoperimetric inequalities in high-dimensional convex sets (2406.01324 - Klartag et al., 3 Jun 2024) in Section 7 (Further localization results), subsection “Life before improved Lichnerowicz”