Tomasz Tkocz
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Stability of Simplex Slicing Open
We establish dimension-free stability of Webb’s sharp simplex slicing (1996). Incidentally, we show the Lipschitzness of volume of central sections of arbitrary (not necessarily symmetric) convex bodies.
Khinchin inequalities for uniforms on spheres with a deficit Open
We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).
From simplex slicing to sharp reverse Hölder inequalities Open
Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave …
Haagerup’s phase transition at polydiscslicing Open
We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres.This provides a probabilistic extension of …
A Rényi entropy interpretation of anti-concentration and noncentral sections of convex bodies Open
We extend Bobkov and Chistyakov's (2015) upper bounds on concentration functions of sums of independent random variables to a multivariate entropic setting. The approach is based on pointwise estimates on densities of sums of independent r…
Stability of simplex slicing Open
We establish dimension-free stability of Webb's sharp simplex slicing (1996). Incidentally, we investigate Lipschitzness of volume of hyperplane central sections of arbitrary (not necessarily symmetric) convex bodies.
Convexity properties of sections of 1-symmetric bodies and Rademacher sums Open
We establish a monotonicity-type property of volume of central hyperplane sections of the 1-symmetric convex bodies, with applications to chessboard cutting. We parallel this for projections with a new convexity-type property for Rademache…
View article: Stability of polydisc slicing
Stability of polydisc slicing Open
We prove a dimension‐free stability result for polydisc slicing due to Oleszkiewicz and Pełczyński. Intriguingly, compared to the real case, there is an additional asymptotic maximizer. In addition to Fourier‐analytic bounds, we crucially …
A Sharp Gaussian Tail Bound for Sums of Uniforms Open
We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.
Stability of polydisc slicing Open
We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pelczyński (2000). Intriguingly, compared to the real case, there is an additional asymptotic maximiser. In addition to Fourier-analytic bounds, we cru…
Distributional stability of the Szarek and Ball inequalities Open
We prove an extension of Szarek's optimal Khinchin inequality (1976) for distributions close to the Rademacher one, when all the weights are uniformly bounded by a $1/\sqrt2$ fraction of their total $\ell_2$-mass. We also show a similar ex…
Typical Values of Extremal-Weight Combinatorial Structures with Independent Symmetric Weights Open
Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisati…
Sharp Rosenthal‐type inequalities for mixtures and log‐concave variables Open
We obtain Rosenthal‐type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremizers in log‐concave settings when the moments of summands ar…
Typical values of extremal-weight combinatorial structures with independent symmetric weights Open
Suppose that the edges of a complete graph are assigned weights independently at random and we ask for the weight of the minimal-weight spanning tree, or perfect matching, or Hamiltonian cycle. For these and several other common optimisati…
Resilience of cube slicing in $\ell_p$ Open
Ball's celebrated cube slicing (1986) asserts that among hyperplane sections of the cube in $\mathbb{R}^n$, the central section orthogonal to $(1,1,0,\dots,0)$ has the greatest volume. We show that the same continues to hold for slicing $\…
Extremal sections and projections of certain convex bodies: a survey Open
We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly $\ell_p$ balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago…
Complex Hanner's Inequality for Many Functions Open
We establish Hanner's inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.
Haagerup's phase transition at polydisc slicing Open
We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of…
Sharp Rosenthal-type inequalities for mixtures and log-concave variables Open
We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands ar…
Entropies of sums of independent gamma random variables Open
We establish several Schur-convexity type results under fixed variance for weighted sums of independent gamma random variables and obtain nonasymptotic bounds on their Rényi entropies. In particular, this pertains to the recent results by …
A Randomly Weighted Minimum Arborescence with a Random Cost Constraint Open
We study the minimum spanning arborescence problem on the complete digraph [Formula: see text], where an edge e has a weight W e and a cost C e , each of which is an independent uniform random variable U s , where [Formula: see text] and U…
Khinchin-Type Inequalities via Hadamard’s Factorisation Open
We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamard’s factorisation theorem from complex analysis, combined with Newton’s inequalities for elementary symmetri…
Tail bounds for sums of independent two-sided exponential random variables Open
We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson's recent results for one-sided exponentials.
Slicing $\ell_p$-balls reloaded: stability, planar sections in $\ell_1$ Open
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…
Improved bounds for Hadwiger’s covering problem via thin-shell estimates Open
A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number N(n) is such that every convex body in \mathbb{R}^{n} can be covered by a union of the interiors of at most N(n) of its tra…
Rademacher–Gaussian tail comparison for complex coefficients and related problems Open
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…