arXiv subjects

Grigory Ivanov

Publications and source records attributed to Grigory Ivanov.

At least 19 recordsLinked to original sources

arXiv2026

The maximal volume of projections of the cross-polytope

We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\dots,v_n \in \mathbb{R}^k, $ we prove \[ \operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}. \] After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every radial simplex with the Gaussian solid angle of its positive cone, and then use that the radial cones form a complete fan. As a consequence, the volume of the projection of $\crosp^n$ onto any $k$-dimensional subspace is at most $2^k/k!$, with equality only for coordinate subspaces.

math.FA↗

arXiv2026

The VC dimension of partial concept classes via Radon's theorem

Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions $f: V\to\{0,1,\ast\}$; equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$). Its VC dimension is defined by shattering sets on which the value $\ast$ is not taken. We study two geometric PCCs in real Banach spaces, both with a margin $\delta>0$: expanded half-spaces, where the grey part is a strip of width at least $\delta$ adjacent to a half-space, and expanded balls, where the grey part is an annulus of width $\delta$ around a unit radius ball. Our main results are dimension-free upper bounds on the VC dimension of the PCC of expanded balls in $L_p\parenth{\mu}$, $1\le p<\infty$, including the non-Euclidean and algorithmically particularly relevant case $\ell^d_1$. These bounds depend on the margin and on the radii, but not on the ambient dimension or the underlying measure space. These are extensions of the work of Bourneuf, Charbit, and Thomass\'e (FOCS 2025) who studied the PCC of expanded balls in Euclidean space, that is, $\ell_2^d$. We also prove lower bounds on the VC dimension that match the upper bounds in terms of the margin parameter $\delta$. Finally, we derive a Dense Neighborhood Lemma in $L_p$-spaces, again extending the known Euclidean results. Our method relies on the linearization of the distance through a map into a space of non-trivial Rademacher type, and then the use of a balanced signed-sum estimate, or a no-dimensional Radon theorem. The arguments rely on ideas from functional analysis that are clearly explained for the non-expert in that field.

cs.LG↗

arXiv2026

Optimality of no-dimensional bounds in Banach spaces

We discuss lower-bound constructions for several no-dimensional theorems of combinatorial geometry in Banach spaces. The common mechanism is the Maurey--Pisier theorem: the supremal Rademacher type of a Banach space forces finite-dimensional $\ell_p$-structures, and standard-coordinate configurations in these model spaces give lower bounds for the error terms. For the Helly approximation property the relevant type is the type of the dual space. For colorful Radon, colorful Tverberg, selection, and weak $\varepsilon$-net statements the relevant type is the type of the original space. We show that the powers appearing in the no-dimensional Helly, Radon, Tverberg, and selection estimates are optimal at the supremal-type exponent. If the supremal type is attained, the known upper estimates coming from the corresponding type inequalities have the best possible order. We also include endpoint statements for spaces of trivial type. In this case the error terms in the Helly, Radon, Tverberg, and selection statements cannot tend to zero. Finally, we prove an endpoint obstruction for no-dimensional weak $\eps$-nets in spaces of trivial type. For every fixed cardinality bound, one can find a finite set in the unit ball for which no approximate weak $\eps$-net of that size exists below a fixed positive radius. The proof combines the simplex example in $\ell_1^N$, the Lov\'asz theorem on the chromatic number of Kneser's graph, and finite representability of $\ell_1^N$ in spaces of trivial type.

math.FA↗

arXiv2026

Greedy sparsifications of sums of positive semidefinite matrices

We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\dots,A_m$ be positive semidefinite $d\times d$ matrices, and let $\lambda_1,\dots,\lambda_m \ge 0$ satisfy \[ \sum_{i=1}^m \lambda_i = 1, \qquad \sum_{i=1}^m \lambda_i A_i = I_d, \qquad \|A_i\| \le M \quad\text{for all } i=1,\dots,m. \] We show that there exists a deterministic sequence of indices $i_1,i_2,\dots \in \{1,\dots,m\}$ such that for every integer $k \ge 1$, \[ \left\| \frac{1}{k}\sum_{r=1}^k A_{i_r} - I_d \right\| \le \begin{cases} \displaystyle \frac{2M\ln(2d)}{k}, & \text{if } k \le M\ln(2d),\\[2ex] \displaystyle 3\sqrt{\frac{M\ln(2d)}{k}}, & \text{if } k > M\ln(2d). \end{cases} \] In particular, if $0<\varepsilon\le 1$ and $N \ge 9M\ln(2d)\varepsilon^{-2}$, then one can choose indices $i_1,\dots,i_N \in \{1,\dots,m\}$ such that \[ \left\| \frac{1}{N}\sum_{r=1}^N A_{i_r} - I_d \right\| \le \varepsilon. \]

math.FA↗

arXiv2026

On Banach Spaces with the Helly Approximation Property

Qualitatively, a no-dimensional Helly-type theorem says that if every small subfamily of convex sets has a common point in a bounded region, then suitable neighborhoods of all the sets in the whole family have a common point. Quantitative bounds, when available, depend on the ambient metric. We say that a Banach space has the Helly approximation property if the radii of these neighborhoods tend to zero as the size of the subfamilies tends to infinity. In this paper, we show that the Helly approximation property holds if and only if the dual space has non-trivial Rademacher type. The argument combines Maurey's empirical method with a duality argument at a minimizer of the maximal distance function. We also prove a colorful version of this theorem, with control over the average of the radii.

math.FA↗

arXiv2026

No-dimensional results of combinatorial convexity. Dimension strikes back

We discuss no-dimensional (approximate) versions of Carath\'eodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several questions. We also point out a simple way to ``bring the dimension back into the picture'': by combining no-dimensional statements with dimension-dependent norm comparisons, one can transfer problems in $\ell_1^d$, $\ell_\infty^d$, and Schatten classes $S_1, S_\infty$ to nearby $\ell_p^d$ or $S_p$ spaces with better geometry. As elementary applications, we obtain a weak additive analogue of the Johnson--Lindenstrauss flattening lemma, local-to-global estimates for Chebyshev regression over the $\ell_1$ ball, and a local-to-global guarantee for quantum feasibility from locally consistent linear measurements.

math.FA↗

arXiv2025

Tight colorful no-dimensional Tverberg theorem

We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius $R$ such that, for any pairwise disjoint $k$-element subsets $Q_1,\dots,Q_n$ of a normed space, there exists a partition of $Q_1\cup\cdots\cup Q_n$ into disjoint transversals $\{P_1,\dots,P_k\}$ for which a ball of radius $R$ intersects the convex hull of each $P_i$ ($1\le i\le k$). Our methods are deterministic and dimension-free, and they are unified by optimizing two functionals: a quadratic \emph{selection} functional whose local maximizers produce a complete system of disjoint transversals, and a convex \emph{intersection} functional that certifies a common point. First, in the Euclidean setting we bound $R$ in terms of the Chebyshev radii (minimal enclosing-ball radii) of the color classes $Q_1,\dots,Q_n$. A key observation is a ``combinatorial'' subadditivity of the squared Chebyshev radius: given sequences $X=(x_1,\dots,x_k)$ and $Y=(y_1,\dots,y_k)$ of points in a Euclidean space, contained in balls of radii $R_X$ and $R_Y$ (not necessarily with the same center), one can reenumerate $Y$ so that the pointwise-sum sequence $Z=(x_1+y_1,\dots,x_k+y_k)$ is contained in a ball of radius $R_Z$ satisfying \[ R_Z^2 \le R_X^2 + R_Y^2 . \] As a corollary, we obtain the best-possible bound \[ R \le \frac{1}{\sqrt{2n}}\sqrt{\frac{k-1}{k}}\, \max_{1\le i\le n} \operatorname{diam}(Q_i). \] Our algorithm returns the desired disjoint transversals in overall time $\mathcal{O}(nk^3)$. Second, we develop a complementary approach based on the inter-color diameter and extend the framework to obtain no-dimensional colorful Tverberg-type results in the hyperbolic setting and in Banach spaces.

math.MG↗

arXiv2025

John Ellipsoids of Revolution

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider scaled copies of the origin-centered closed unit ball, whereas in the second problem, we search among all affine copies of the unit ball. In this paper, we interpolate between these two classical problems by considering ellipsoids of revolution. More generally, we study pairs of convex bodies $K$ and $L$, and seek a largest-volume affine image of $K$ contained within $L$, subject to certain restrictions on the allowed affine transformations. We derive first-order necessary conditions for optimality, generalizing known conditions from the unrestricted affine setting. Using these conditions, we show that an extremal ellipsoid of revolution exhibits properties analogous to those of either the largest-volume ellipsoid or the largest Euclidean ball, depending on whether the ellipsoid is considered along its axis of revolution or along the orthogonal complement of that axis.

math.MG↗

arXiv2025

Quantitative Steinitz theorem and polarity

The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior. Bárány, Katchalski, and Pach established a quantitative version of Steinitz's theorem, showing that for a convex polytope $Q$ in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$, there exist at most $2d$ vertices of $Q$ whose convex hull $Q'$ satisfies $r\mathbf{B}^d \subset Q' $ with $r \geq d^{-2d}$. Recently, Márton Naszódi and the author derived a polynomial bound on $r$. This paper aims to establish a bound on $r$ based on the number of vertices of $Q.$ In other words, we demonstrate an effective method to remove several points from the original set $Q$ without significantly altering the bound on $r$. Specifically, if the number of vertices of $Q$ scales linearly with the dimension, i.e., $αd$, then one can select $2d$ vertices such that $r \geq \frac{1}{5 αd}$. The proof relies on a polarity trick, which may be of independent interest: we demonstrate the existence of a point $c$ in the interior of a convex polytope $P \subset \mathbb{R}^d$ such that the vertices of the polar polytope $(P-c)^\circ$ sum up to zero.

math.MG↗

arXiv2024

Colorful positive bases decomposition and Helly-type results for cones

We prove the following colorful Helly-type result: Fix $k \in [d-1]$. Assume $\mathcal{A}_1, \dots, \mathcal{A}_{d+(d-k)+1}$ are finite sets (colors) of nonzero vectors in $\R^d$. If for every rainbow sub-selection $R$ from these sets of size at most $\max \{d+1, 2(d-k+1)\}$, the system $\langle {a},{x} \rangle \leq 0,\; a \in R$ has at least $k$ linearly independent solutions, then at least one of the systems $\langle {a},{x} \rangle \leq 0,\; a \in \mathcal{A}_i,$ $i \in [d+(d-k)+1]$ has at least $k$ linearly independent solutions. A \emph{rainbow sub-selection} from several sets refers to choosing at most one element from each set (color). The Helly number $\max \{d+1, 2(d-k+1)\}$ and the number of colors $d+(d-k)+1$ are optimal. Our key observation is a certain colorful Carath\'eodory-type result for positive bases.

math.CO↗

arXiv2023

Quantitative Steinitz theorem: A spherical version

Steinitz's theorem states that if the origin belongs to the interior of the convex hull of a set $Q \subset \mathbb{R}^d$, then there are at most $2d$ points $Q^\prime$ of $Q$ whose convex hull contains the origin in the interior. Bárány, Katchalski and Pach gave a quantitative version whereby the radius of the ball contained in the convex hull of $Q^\prime$ is bounded from below. In the present note, we show that a Euclidean result of this kind implies a corresponding spherical version.

math.MG↗

arXiv2023

Functional John and Löwner conditions for pairs of log-concave functions

John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body $L$. Another, more recent direction is to consider logarithmically concave functions on $\mathbb{R}^d$ instead of convex bodies: we designate some special, radially symmetric log-concave function $g$ as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function $f$. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of $g$ above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called \emph{L{ö}wner's ellipsoid}, containing $K$. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function $g$ under the constraint that it is pointwise above $f$. It turns out that in the functional setting, the relationship between the John and the L{ö}wner problems is more intricate than it is in the setting of convex bodies.

math.MG↗

arXiv2023

Quantitative Steinitz Theorem: A polynomial bound

The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let $Q$ be a convex polytope in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q^\prime$ satisfies \[ r \mathbf{B}^d \subset Q^\prime \] with $r\geq d^{-2d}$. They conjectured that $r\geq c d^{-1/2}$ holds with a universal constant $c>0$. We prove $r \geq \frac{1}{5d^2}$, the first polynomial lower bound on $r$. Furthermore, we show that $r$ is not be greater than $\frac{2}{\sqrt{d}}$.

math.MG↗

arXiv2023

Geometric representation of classes of concave functions and duality

Using a natural representation of a $1/s$-concave function on $\mathbb{R}^d$ as a convex set in $\mathbb{R}^{d+1},$ we derive a simple formula for the integral of its $s$-polar. This leads to convexity properties of the integral of the $s$-polar function with respect to the center of polarity. In particular, we prove that that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for $s$-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.

math.FA↗

arXiv2022

Erdős--Ko--Rado and Hilton--Milner theorems for two-forms

In this short note we show that both generalizations of celebrated Erdős--Ko--Rado theorem and Hilton--Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from the folklore puzzle about possible arrangements of an intersecting family of lines.

math.CO↗

arXiv2021

Functional John Ellipsoids

We introduce a new way of representing logarithmically concave functions on $\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every $s>0$, we define a class of non-negative functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$. For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to $f$, and we call it the \emph{\jsfunction} of $f$. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John $s$-functions converge to characteristic functions of ellipsoids as $s$ tends to zero and to Gaussian densities as $s$ tends to infinity. As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant $c_d$ multiple of the integral of the pointwise minimum of a properly chosen subfamily of size $3d+2$, where $c_d$ depends only on $d$.

math.FA↗

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.

math.MG↗

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.

math.FA↗