Optimal pointwise lower bounds for even log-concave densities

Speaker(s)

Daniel Murawski

Affiliation

University of Warsaw

Language of the talk

English

Date

Jan. 22, 2026, 12:15 p.m.

Room

room 3160

Title in Polish

Optimal pointwise lower bounds for even log-concave densities

Seminar

Seminar of Probability Group

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We show that among all symmetric real log-concave random variables X with variance 1 and any t_0 in range [0, √3] the quantity f_X(t_0) is minimized by a uniform, Laplace or truncated Laplace distribution. We show that for t_0 ≥ 1/√2 the minimum is attained by Laplace distribution and for t_0 ≤ 1/2 it is attained by uniform distribution. We also show that the constant 1/√2 cannot be improved and that there exist t_0 such that the minimizer is neither Laplace or uniform. This gives optimal dimension-free lower bounds for measures of non-central slices of isotropic convex bodies.

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