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Roman Karasev

Publications and source records attributed to Roman Karasev.

59 records · Page 4Linked to original sources
arXiv2012

Extensions of theorems of Rattray and Makeev

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously, when we consider orthonormal $k$-frames in $\R^n$ instead of orthonormal basis (full frames). We also present new results on simultaneous partition of several measures into parts by $k$ mutually orthogonal hyperplanes. In the case $k=2$ we relate the Rattray and Makeev type results with the well known embedding problem for projective spaces.

math.AT↗
arXiv2012

Estimating the higher symmetric topological complexity of spheres

We study questions of the following type: Can one assign continuously and $Σ_m$-equivariantly to any $m$-tuple of distinct points on the sphere $S^n$ a multipath in $S^n$ spanning these points? A \emph{multipath} is a continuous map of the wedge of $m$ segments to the sphere. This question is connected with the \emph{higher symmetric topological complexity} of spheres, introduced and studied by I. Basabe, J. González, Yu. B. Rudyak, and D. Tamaki. In all cases we can handle, the answer is negative. Our arguments are in the spirit of the definition of the Hopf invariant of a map $f: S^{2n-1} \to S^n$ by means of the mapping cone and the cup product.

math.AT↗
arXiv2011

Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carathéodory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg type theorem for families of convex compacta.

math.MG↗
arXiv2011

Kadets type theorems for partitions of a convex body

For convex partitions of a convex body $B$ we prove that we can put a homothetic copy of $B$ into each set of the partition so that the sum of homothety coefficients is $\ge 1$. In the plane the partition may be arbitrary, while in higher dimensions we need certain restrictions on the partition.

math.CO↗