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math.PR2026

Cramér transform, half-space depth and threshold phenomena for convex bodies

Minas Pafis

We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure on a convex body…

math.PR2026

Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes

Minas Pafis, Christos Pandis

We establish variance lower bounds for the intrinsic volumes and the face numbers of random polytopes, generated by independent samples from rotationally invariant log-concave prob…

math.PR2026

Many Facets in Random Polytopes from Product and Log-Concave Measures

Silouanos Brazitikos, Minas Pafis

We prove bounds of order for the expected number of facets of high-dimensional random polytopes. First, let be a non-degenerate compactly supported even proba…

math.PR2026

Discrete log-concavity and threshold phenomena for atomic measures

Silouanos Brazitikos, Minas Pafis

We investigate threshold phenomena for random polytopes $K_N=\conv\{X_1,\dots,X_N\}$ generated by i.i.d.\ samples from an atomic law . We identify and provide a missing justific…