arXiv subjects
Publications and source records attributed to R. N. Karasev.
In this paper some results on the topology of the space of $k$-flats in $\mathbb R^n$ are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in $\mathbb R^n$, and on measure partitions by hyperplanes, are deduced.
We prove some analogues of the central point theorem and Tverberg's theorem, where instead of points, we consider hyperplanes or affine flats of given dimension.
In this paper a version of Knaster-Kuratowski-Mazurkiewicz theorem for products of simplices is formulated. Some corollaries for measure partition in the plane and cutting families of sets in the plane by lines are given.
We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans.
In this paper we prove that it is possible to inscribe a regular crosspolytope (multidimensional octahedron) into a smooth convex body in $\mathbb R^d$, where $d$ is an odd prime power. Some generalizations of this statement are also proved.
In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly $p$-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii-Schwarz, and the equivariant Lyusternik-Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced.
In this paper some new cases of Knaster's problem on continuous maps from spheres are established. In particular, we consider an almost orbit of a $p$-torus $X$ on the sphere, a continuous map $f$ from the sphere to the real line or real plane, and show that $X$ can be rotated so that $f$ becomes constant on $X$.
We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we obtain the lower bound $(d-2)(p-1)+2$, which is much better than the previous estimates.