arXiv subjects

Grigory Ivanov

Publications and source records attributed to Grigory Ivanov.

30 records · Page 2Linked to original sources

arXiv2021

A Quantitative Helly-type Theorem: Containment in a Homothet

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the intersection of finitely many convex bodies in $\mathbb{R}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d)^3\mathrm{diam}(K)$. The best previously known estimate, due to Brazitikos, is $c d^{11/2}$. Moreover, we confirm that the multiplicative factor $c d^{1/2}$ conjectured by Bárány, Katchalski and Pach cannot be improved.

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arXiv2020

Rectifiable curves in proximally smooth sets

We provide an algorithm of constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space. Our algorithm returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses a certain modification of the metric projection. We estimate the length of a constructed curve and its deviation from the segment with the same endpoints. These estimates coincide up to a constant factor with those for the geodesics in a proximally smooth set in a Hilbert space.

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arXiv2020

Functional Löwner Ellipsoids

We extend the notion of the smallest volume ellipsoid containing a convex body in~$\mathbb{R}^{d}$ to the setting of logarithmically concave functions. We consider a vast class of logarithmically concave functions whose superlevel sets are concentric ellipsoids. For a fixed function from this class, we consider the set of all its "affine" positions. For any log-concave function $f$ on $\mathbb{R}^{d},$ we consider functions belonging to this set of "affine" positions, and find the one with the smallest integral under the condition that it is pointwise greater than or equal to $f.$ We study the properties of existence and uniqueness of the solution to this problem. For any $s \in [0,\infty),$ we consider the construction dual to the recently defined John $s$-function \cite{ivanov2020functional}. We prove that such a construction determines a unique function and call it the \emph{Löwner $s$-function} of $f.$ We study the Löwner $s$-functions as $s$ tends to zero and to infinity. Finally, extending the notion of the outer volume ratio, we define the outer integral ratio of a log-concave function and give an asymptotically tight bound on it. \end{abstract}

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arXiv2020

Approximation of the average of some random matrices

Rudelson's theorem states that if for a set of unit vectors $u_i$ and positive weights $c_i$, we have that $\sum c_i u_i\otimes u_i$ is the identity operator $I$ on ${\mathbb R}^d$, then the sum of a random sample of $Cd\ln d$ of these diadic products is close to $I$. The $\ln d$ term cannot be removed. On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the $\ln d$ term can be removed, if one wants to show the existence of a good approximation of $I$ as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semi-definite matrices. First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by $Cd$ elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices. Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach--Mazur distance of convex bodies. Finally, we show that in some cases, one needs to take a subset of the vectors of order $d^2$ to approximate the identity.

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arXiv2020

On the volume of sections of the cube

We study the properties of the maximal volume $k$-dimensional sections of the $n$-dimensional cube $[-1,1]^n$. We obtain a first order necessary condition for a $k$-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of $\mathbb{R}^n$ onto a $k$-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube $[-1,1]^n,$ $n \geq 2.$

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arXiv2019

No-dimension Tverberg's theorem and its corollaries in Banach spaces of type $p$

We continue our study of 'no-dimension' analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper \cite{adiprasito2019theorems} and prove no-dimension versions of colorful Tverberg's theorem, selection lemma and the weak $ε$-net theorem in Banach spaces of type $p > 1.$ To prove this results we use the original ideas of \cite{adiprasito2019theorems} for the Euclidean case and our slightly modified version of the celebrated Maurey lemma.

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arXiv2019

Tight frames and related geometric problems

We study the properties of a set of vectors called tight frames that obtained as the orthogonal projection of some orthonormal basis of $\R^n$ onto $\R^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary maximum condition of these problems. As applications, we prove new results for the problem of the maximization of the volume of zonotopes.

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arXiv2019

Orthodiagonal anti-involutive Kokotsakis polyhedra

We study the properties of Kokotsakis polyhedra of orthodiagonal anti-involutive type. Stachel conjectured that a certain resultant connected to a polynomial system describing flexion of a Kokotsakis polyhedron must be reducible. Izmestiev \cite{izmestiev2016classification} showed that a polyhedron of the orthodiagonal anti-involutive type is the only possible candidate to disprove Stachel's conjecture. We show that the corresponding resultant is reducible, thereby confirming the conjecture. We do it in two ways: by factorization of the corresponding resultant and providing a simple geometric proof. We describe the space of parameters for which such a polyhedron exists and show that this space is non-empty. We show that a Kokotsakis polyhedron of orthodiagonal anti-involutive type is flexible and give explicit parameterizations in elementary functions and in elliptic functions of its flexion.

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arXiv2017

On the volume of the John-Löwner ellipsoid

We find an optimal upper bound on the volume of the John ellipsoid of a $k$-dimensional section of the $n$-dimensional cube, and an optimal lower bound on the volume of the Löwner ellipsoid of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We use these results to give a new proof of Ball's upper bound on the volume of a $k$-dimensional section of the hypercube, and of Barthe's lower bound on the volume of a projection of the $n$-dimensional cross-polytope onto a $k$-dimensional subspace. We settle equality cases in these inequalities. Also, we describe all possible vectors in $\R^n,$ whose coordinates are the squared lengths of a projection of the standard basis in $\R^n$ onto a $k$-dimensional subspace.

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arXiv2015

Hypomonotonicity of the normal cone and proximal smoothness

In this paper we study the properties of the normal cone to the proximally smooth set. We give the complete characterization of the proximally smooth set through the monotony properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We give the exact bounds for right-hand side in the monotonicity inequality for normal cone in terms of the moduli of smoothness and convexity of a Banach space.

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arXiv2015

Modulus of supporting convexity and supporting smoothness

We introduce the moduli of the supporting convexity and the supporting smoothness of a Banach space, which characterize the deviation of the unit sphere from an arbitrary supporting hyperplane. We show that the modulus of supporting smoothness, the Bana{ś} modulus, and the modulus of smoothness are all equivalent at zero, the modulus of supporting convexity is equivalent at zero to the modulus of convexity. We prove a Day--Nordlander type result for these moduli.

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arXiv2015

Convex hull deviation and contractibility

We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-module of space $X$ as the supremum of this distance for all subset of the unit ball in $X$. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-module depending on the dimension of the space. We give an upper bound for the CHD-module in $L_p$ spaces. We prove that CHD-module is not greater than the maximum of the Lipschitz constants of metric projection operator onto hyperplanes. This implies that for a Hilbert space CHD-module equals 1. We prove criterion of the Hilbert space and study the contractibility of proximally smooth sets in uniformly convex and uniformly smooth Banach spaces.

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