Related papers: The slicing conjecture via small ball estimates
We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order…
This article develops, via the perspective of (arithmetic) algebraic geometry and category theory, different aspects of geometry of information. First, we describe in the terms of Eilenberg--Moore algebras over a Giry monad, the collection…
The Littlewood Conjecture in Diophantine approximation can be thought of as a problem about covering the plane by a union of hyperbolas centered at rational points. In this paper we consider the problem of translating the center of each…
Given a rank 3 real arrangement $\mathcal A$ of $n$ lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for $n$ sufficiently large, the number of simple intersection points of $\mathcal…
The Kannan-Lov\'asz-Simonovits conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and…
We study the $L^p-L^q$ boundedness of Bergman projector on the minimal ball. This improves an important result of \cite{MY} due to G. Mengotti and E. H. Youssfi.
A proof of Thompson's conjecture for real semi-simple Lie groups has been given by Kapovich, Millson, and Leeb. In this note, we give another proof of the conjecture by using a theorem of Alekseev, Meinrenken, and Woodward from symplectic…
This is a comprehensive set of notes on the ArXiV paper math.CA/0609815 by Dmitry Bilyk and the author. The focus of that paper is a new inequality for sums of hyperbolic Haar functions in three variables, extending a famous result of J…
We investigate geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analogue of this geometric inequality…
We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal…
In the simplest case, consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann's first conjecture, the probability that the so…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
An alternative computational approach to the Collatz (3n+1) conjecture is presented that may be theoretically capable of confirming the conjecture.
McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a four-dimensional ball, and found that when the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by…
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case…
In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$…
While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential…
Foulkes' conjecture has several generalisations due to Doran, Abdesselam--Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, these assert that given $a \le c \le d \le b$ with $ab=cd$, the…
The purpose of this short note is to demonstrate how some techniques from additive combinatorics recently developed by Peluse and Peluse-Prendiville can be applied to give an alternative proof for a trilinear smoothing inequality originally…