An isoperimetric inequality for mean shadow
We prove an isoperimetric inequality in terms of the mean shadow area.
math.MG↗
arXiv subjects
Roman Karasev
Publications and source records attributed to Roman Karasev.
The mapping index through the lens of the cross-index
We study the cross-index of free \(G\)-posets as a combinatorial analogue of the equivariant topological index. We demonstrate that the cross-index exhibits many structural properties closely paralleling those of the topological index, while its behavior with respect to unions displays a pronounced dichotomy depending on the acting group. Specifically, if \(P = A \cup B\) is a union of \(G\)-invariant subposets, then for \(G = \mathbb{Z}_2\) we obtain the sharp inequality \[ \operatorname{xind} P \le \operatorname{xind} A + \operatorname{xind} B + 1, \] which is directly analogous to the classical union inequality for the topological index. In contrast, for every group \(G\neq \mathbb{Z}_2\), this phenomenon fails in general, and we establish the best possible weaker estimate \[ \operatorname{xind} P \le \operatorname{xind} A + 2(\operatorname{xind} B+1). \] This reveals a fundamental distinction between the \(\mathbb{Z}_2\)-equivariant and non-\(\mathbb{Z}_2\)-equivariant settings at the purely combinatorial level. As further consequences, we compare the cross-index with both the topological index and the simplicial index, showing in particular that the gap between the cross-index and the topological index can be arbitrarily large. These results clarify the role of the cross-index as a combinatorial analogue of the equivariant topological index and further strengthen the interplay between equivariant topological methods and combinatorial structures endowed with symmetry.
math.CO↗
Tensor rank of the determinant and periodic triangulations of $\mathbb{R}^n$
We prove that in any $\mathbb{Z}^n$-periodic triangulation of $\mathbb{R}^n$ the number of $\mathbb{Z}^n$-orbits of $n$-dimensional simplices is at least the tensor rank of the $n$th determinant tensor. The latter is known to be at least $\frac{n^{n-1}}{(n-1)!}$, which is approximately $\frac{e^n}{\sqrt{2\pi n}}$ for large $n$. The triangulation is not assumed to be geometric, meaning that its simplices can be ``curved''. We also provide lower bounds for general spaces. A simplicial cell complex is a CW-complex glued out of simplices with the attaching maps being simplicial embeddings; this notion generalizes simplicial complexes. We prove that if $X$ is a simplicial cell complex with cohomological classes $\alpha_i\in H^{d_i}(X;\mathbb{Z}_2)$ satisfying \[ \alpha_1 \smile \alpha_2 \smile \ldots \smile \alpha_n \neq 0, \] then $X$ has at least $2^n$ simplices of dimension $d_1+d_2+\ldots+d_n$. In particular, a simplicial cell complex homeomorphic to $\mathbb{R} P^n$, $\mathbb{C} P^n$, or $(S^2)^n$, has at least $2^n$ top-dimensional simplices. A crystallization of a manifold is a simplicial cell complex homeomorphic to this manifold and having the least possible number of vertices. We give a short explicit construction of a crystallization and a triangulation of $\mathbb{R}^n/\mathbb{Z}^n$ with $n+1$ and $2^{n+1}-1$ vertices, resp. Triangulations with this many vertices were described before and no smaller triangulation is known.
math.CO↗
Rogers's proof of Vaaler's theorem
We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.
math.MG↗
Extensions of polynomial plank covering theorems
We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.
math.MG↗
Short proofs of Tverberg-type theorems for cell complexes
We present short proofs of Tverberg-type theorems for cell complexes by S. Hasui, D. Kishimoto, M. Takeda, and M. Tsutaya. One of them states that for any prime power $r$, any complex $X$ topologically homeomorphic to $S^{(d+1)(r-1)-1}$, and any continuous map $f:X\to\mathbb R^d$ there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $X$ such that $f(\sigma_1)\cap\ldots f(\sigma_r)\ne\emptyset$.
math.GT↗
Weak $(1-ε)$-nets for polynomial superlevel sets
We prove that for any Borel probability measure $μ$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x \in X$) satisfies $μ\{P\geq 0\}\geq \frac{2}{(n+1)(n+2)}$. We also prove that given an absolutely continuous probability measure $μ$ on $\mathbb R^n$ and $D\leq 2k$, for every $δ>0$ there exists a set $X\subset \mathbb R^n$ with $|X|\leq \binom{n+2k}{n}-n-1$ such that any $n$-variate polynomial $P$ of degree $D$ that is nonnegative on $X$ satisfies $μ\{P\geq 0\}\geq \frac{1}{\binom{n+2k}{n}+1} - δ$. These statements are analogues of the celebrated centerpoint theorem, which corresponds to the case of linear polynomials. Our results follow from new estimates on the Carathéodory numbers of real Veronese varieties, or alternatively, from bounds on the nonnegative symmetric rank of real symmetric tensors.
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Convex bodies with all characteristics planar
We show that smooth and strongly convex bodies in the symplectic $\mathbb R^{2n}$ for $n>1$ with all characteristics planar, or all outer billiard trajectories planar are affine symplectic images of balls.
math.SG↗
Symplectic polarity and Mahler's conjecture
We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds for symplectic capacities of symplectically self-polar bodies.
math.MG↗
Systolic inequalities and chromatic number
We show that the discrete versions of the systolic inequality that estimate the number of vertices of a simplicial complex from below have substantial applications to graphs, the one-dimensional simplicial complexes. Almost directly they provide good estimates for the number of vertices of a graph in terms of its chromatic number and the length of the smallest odd cycle. Combined with the graph-theoretic techniques of Berlov and Bogdanov, the systolic approach produces even better estimates.
math.CO↗
Covering by planks and avoiding zeros of polynomials
We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, there exists a point in the unit $d$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $P$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $π$.
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Systolic inequalities for the number of vertices
Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.
math.MG↗
Equipartition of a segment
We prove that, for any positive integer $m$, a segment may be partitioned into $m$ possibly degenerate or empty segments with equal values of a continuous function $f$ of a segment, assuming that $f$ may take positive and negative values, but its value on degenerate or empty segments is zero.
math.MG↗
Topological lower bounds on the sizes of simplicial complexes and simplicial sets
We prove that if an $n$-dimensional space $X$ satisfies certain topological conditions then any triangulation of $X$ as well as any its representation as a simplicial set with contractible faces has at least $2^n$ faces of dimension $n$. One example of such $X$ is the $n$-dimensional torus $(S^1)^n$.
math.AT↗
A subexponential size triangulation of $\mathbb{R}P^n$
We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $\mathbb{RP}^n$ of size $e^{(\frac{1}{2}+o(1))\sqrt{n}{\log n}}$.
math.CO↗
Envy-free division using mapping degree
In this paper we study envy-free division problems. The classical approach to such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem). We follow Erel Segal-Halevi, Frédéric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free division problems when $n$ is odd and not a prime power. In this arxiv version that appears after the official publication we have corrected the statement and the proof of Lemma 3.4.
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More bisections by hyperplane arrangements
A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojević, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grünbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.
math.MG↗
Mahler's conjecture for some hyperplane sections
We use symplectic techniques to obtain partial results on Mahler's conjecture about the product of the volume of a convex body and the volume of its polar. We confirm the conjecture for hyperplane sections or projections of $\ell_p$-balls or the Hanner polytopes.
math.MG↗