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Roman Karasev

Publications and source records attributed to Roman Karasev.

At least 55 records · Page 3Linked to original sources
arXiv2016

Colorful theorems for strong convexity

We prove two colorful Carathéodory theorems for strongly convex hulls, generalizing the colorful Caratéodory theorem for ordinary convexity by Imre Bárány, the non-colorful Carathéodory theorem for strongly convex hulls by the second author, and the "very colorful theorems" by the first author and others. We also investigate if the assumption of a "generating convex set" is really needed in such results and try to give a topological criterion for one convex body to be a Minkowski summand of another.

math.CO↗
arXiv2016

Bounding minimal solid angles of polytopes

In this article we study the following question: What can be the measure of the minimal solid angle of a simplex in $\mathbb{R}^d$? We show that in dimensions three it is not greater than the solid angle of the regular simplex. And in dimension four the same holds for simplices sufficiently close to the regular simplex. We also study a similar question for trihedral and dihedral angles of polyhedra compared to those of regular solids.

math.MG↗
arXiv2016

Local multiplicity of continuous maps between manifolds

Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f\colon M\longrightarrow N$ admits a local $k$-multiplicity if, for every real number $ω>0$, there exist $k$ pairwise distinct points $x_1,\ldots,x_k$ in $M$ such that $f(x_1)=\cdots=f(x_k)$ and $\diam\{x_1,\ldots,x_k\}<ω$. In this paper we systematically study the existence of local $k$-mutiplicities and derive criteria for the existence of local $k$-multiplicity in terms of Stiefel--Whitney classes and Chern classes of the vector bundle $f^*τN\oplus(-τM)$. For example, as a corollary of one criterion we deduce that for $k\geq 2$ a power of $2$, $M$ a compact smooth manifold with the integer $s:=\max\{\ell : \bar{w}_{\ell}(M)\neq 0\}$, and $N$ a parallelizable smooth manifold, if $s\geq \dim N-\dim M+1$ and $\bar{w}_{s}(M)^{k-1}\neq 0$, any continuous map $M\longrightarrow N$ admits a local $k$-multiplicity. Furthermore, as a special case of this corollary we recover, when $k=2$, the classical criterion for the non-existence of an immersion $M\looparrowright N$ between manifolds $M$ and $N$.

math.AT↗
arXiv2016

Dependence of the heavily covered point on parameters

We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. We have no continuous dependence, but manage to utilize the "homological continuous dependence" of the heavily covered point. This allows us to infer some corollaries in a usual way. We also give an elementary argument to prove the simplest of these corollaries.

math.MG↗
arXiv2015

Bounds for Pach's selection theorem and for the minimum solid angle in a simplex

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$ such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of $\mathbb{R}^d$, then we can find a point $\mathbf{p} \in \mathbb{R}^d$ and subsets $Y_i \subseteq X_i$ for every $i \in [d+1]$, each of size at least $c_d n$, such that $\mathbf{p}$ belongs to all {\em rainbow} $d$-simplices determined by $Y_1,..., Y_{d+1}$, that is, simplices with one vertex in each $Y_i$. We show a super-exponentially decreasing upper bound $c_d\leq e^{-(1/2-o(1))(d \ln d)}$. The ideas used in the proof of the upper bound also help us prove Pach's theorem with $c_d \geq 2^{-2^{d^2 + O(d)}}$, which is a lower bound doubly exponentially decreasing in $d$ (up to some polynomial in the exponent). For comparison, Pach's original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach's theorem with $c_d \geq2^{-O(d^2\log d)}$. In our construction for the upper bound, we use the fact that the minimum solid angle of every $d$-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the "separation" part of the argument by showing that in one of the key steps only $d+1$ separations are necessary, compared to $2^d$ separations in the original proof. We also provide a measure version of Pach's theorem.

math.MG↗
arXiv2015

Topology of geometric joins

We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carathéodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is contractible. We are able to prove this when $d$ equals $2$ and $3$, while for larger $d$ we show that the geometric join is contractible provided the number of sets is quadratic in $d$. We also consider a matroid generalization of geometric joins and provide similar bounds in this case.

math.MG↗
arXiv2014

Measure Partitions Using Hyperplanes with Fixed Directions

We study nested partitions of $R^d$ obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among $k$ sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any $t$ measures there is a path formed only by horizontal and vertical segments using at most $t-1$ turns that splits them by half simultaneously, and optimal mass-partitioning results for chessboard-colourings of $R^d$ using hyperplanes with fixed directions.

math.MG↗
arXiv2013

Another ham sandwich in the plane

We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from arbitrary k-fan when k is odd and in some cases when k is even. We also give some counterexamples for certain fans and measures.

math.MG↗
arXiv2013

From Symplectic Measurements to the Mahler Conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

math.MG↗
arXiv2013

Suborbits in Knaster's problem

In this paper we exhibit a similarity between Euclidean Ramsey problems and Knaster-type problems. By borrowing ideas from Ramsey theory we prove weak Knaster properties of non-equatorial triangles in spheres, and of simplices in Euclidean spaces.

math.FA↗
arXiv2013

Covering dimension using toric varieties

In this paper we deduce the Lebesgue and the Knaster--Kuratowski--Mazurkiewicz theorems on the covering dimension, as well as their certain generalizations, from some simple facts of toric geometry. This provides a new point of view on this circle of results.

math.MG↗
arXiv2012

The Schwarz genus of the Stiefel manifold and counting geometric configurations

In this paper we compute: the Schwarz genus of the Stiefel manifold $V_k(\mathbb R^n)$ with respect to the action of the Weyl group $W_k:=(\mathbb Z/2)^{k}\rtimesΣ_k$, and the Lusternik--Schnirelmann category of the quotient space $V_k(\mathbb R^n)/W_k$. Furthermore, these results are used in estimating the number of: critically outscribed parallelotopes around the strictly convex body, and Birkhoff--James orthogonal bases of the normed finite dimensional vector space.

math.AT↗
arXiv2012

Projective center point and Tverberg theorems

We present projective versions of the center point theorem and Tverberg's theorem, interpolating between the original and the so-called "dual" center point and Tverberg theorems. Furthermore we give a common generalization of these and many other known (transversal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as some essentially new results about partitioning measures in projective space.

math.AT↗
arXiv2012

Turán numbers for $K_{s,t}$-free graphs: topological obstructions and algebraic constructions

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.

math.CO↗
arXiv2012

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.

math.MG↗