Generalized Gr\"unbaum inequality
M. Meyer, F. Nazarov, D. Ryabogin, and V. Yaskin

TL;DR
This paper extends the Grünbaum inequality to integrable log-concave functions in n-dimensional space, establishing a sharp lower bound for integrals over rays emanating from the origin in any direction.
Contribution
It generalizes the Grünbaum inequality from convex bodies to log-concave functions, providing the best possible constant in this broader setting.
Findings
Established a lower bound for integrals of log-concave functions along rays.
Proved the constant e^{-n} is optimal for the inequality.
Extended classical geometric inequalities to a functional setting.
Abstract
Let be an integrable log-concave function on with the center of mass at the origin. We show that for every , and the constant is the best possible.
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Taxonomy
TopicsPoint processes and geometric inequalities · Mathematics and Applications · Mathematical Inequalities and Applications
