Random sections of star bodies
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This thesis concerns analytic and geometric aspects of random sections of star bodies and their implications for problems in stochastic geometry. We treat isoperimetric inequalities and distribution of volumes of such sections. In particular, we resolve a longstanding conjecture on affine isoperimetric inequalities in Lp spaces. The first part of this work is dedicated to the study of isoperimetric inequalities of dual centroid bodies. This family interpolates between the Busemann intersection inequality and the Lutwak-Zhang inequalities, two fundamental affine isoperimetric principles. Consequently, we establish new bridges between Brunn-Minkowski and dual Brunn-Minkowski theories that handle both through a uniform treatment. The approach relies on a new family of random star bodies. The second part concerns limit theorem for sections of star bodies. We establish Central Limit Theorem (CLT) for the volumes of intersections of Bnp with uniform random subspaces of codimension d for fixed d and n --> [infinity]. We refine previous estimates for expected volumes of Koldobksy and Lifshits for 0 < p < 1 and the approximation obtained by the Eldan-Klartag version of CLT for general convex bodies when 1 [less than or equal to] p < 2. In the last part, we provide a short alternative proof of the Busemann intersection inequality. The approach depends on a randomized approximation inspired by a construction of Anttila, Ball, and Perissinaki.
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Ph. D.