We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $ell_p$-balls in all the cases where the extremi
sers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.
We provide a generalisation of Pinelis Rademacher-Gaussian tail comparison to complex coefficients. We also establish uniform bounds on the probability that the magnitude of weighted sums of independent random vectors uniform on Euclidean spheres with matrix coefficients exceeds its second moment.
We provide a sharp lower bound on the $p$-norm of a sum of independent uniform random variables in terms of its variance when $0 < p < 1$. We address an analogous question for $p$-Renyi entropy for $p$ in the same range.
We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamards factorisation theorem from complex analysis, combined with Newtons inequalities for elementary symmetric functions.
Besides the case of independent summands, we also treat ferromagnetic dependencies in a nonnegative external magnetic field (thanks to Newmans generalisation of the Lee-Yang theorem). Lastly, we compare the notions of type L, ultra sub-Gaussianity (introduced by Nayar and Oleszkiewicz) and strong log-concavity (introduced by Gurvits), with the latter two being equivalent.
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Balls cube slicing inequality.
We consider cost constrain
We consider a constrained version of the shortest path problem on the complete graphs whose edges have independent random lengths and costs. We establish the asymptotic value of the minimum length as a function of the cost-budget within a wide range.
We establish a discrete analog of the Renyi entropy comparison due to Bobkov and Madiman. For log-concave variables on the integers, the min entropy is within log e of the usual Shannon entropy. Additionally we investigate the entropic Rogers-Shephar
d inequality studied by Madiman and Kontoyannis, and establish a sharp Renyi version for certain parameters in both the continuous and discrete cases
We study the expected volume of random polytopes generated by taking the convex hull of independent identically distributed points from a given distribution. We show that for log-concave distributions supported on convex bodies, we need at least expo
nentially many (in dimension) samples for the expected volume to be significant and that super-exponentially many samples suffice for concave measures when their parameter of concavity is positive.
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-
Furedi-McDiarmid that exponentially many samples suffice when the convex body is the hypercube, and by Pivovarov that the Euclidean ball demands roughly $d^{d/2}$ samples. We show that when the convex body is the simplex, exponentially many samples suffice; this then implies the same result for any convex simplicial polytope with at most exponentially many faces.
