Related papers: On Borsuk-Ulam theorems and convex sets

The Borsuk-Ulam theorem states that a continuous function $f:S^n \to \R^n$ has a point $x\in S^n$ with $f(x)=f(-x)$. We give an analogue of this theorem for digital images, which are modeled as discrete spaces of adjacent pixels equipped…

Suppose that $f_1,\ldots ,f_m : S(V)\to R$ are $m$ ($\geq 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam…

Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $\phi:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $\lambda\neq 1$ be…

We describe a connective $K$-theory Borsuk--Ulam/Bourgin--Yang theorem for cyclic groups of order a power of a prime $p$. Consider two finite dimensional complex representations $U$ and $V$ of the cyclic group $Z /p^{k+1}$ of order…

Borsuk-Ulam's theorem is a useful tool of algebraic topology. It states that for any continuous mapping $f$ from the $n$-sphere to the $n$-dimensional Euclidean space, there exists a pair of antipodal points such that $f(x)=f(-x)$. As for…

B\'ar\'any's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $\Delta_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $\Delta_N$ to the same point in…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

Let $G$ be a compact Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere…

In order to investigate minimal sufficient conditions for an abstract integral to belong to the convex hull of the integrand, we propose a system of axioms under which it happens. If the integrand is a continuous $R^n$-valued function over…

Let $(\mathcal{M}, c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in (\mathcal{M}, c_k,n_k)$ is $[0,1]$. Let $A,B$ be two non-empty sets in $\mathbb{R}$. Suppose that $f$ is a continuous function defined on an…

We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such…

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of…

We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of…

Let $(\Omega,{\cal F},P)$ be a probability space and $L^{0}({\cal F},R)$ the algebra of equivalence classes of real-valued random variables on $(\Omega,{\cal F},P)$. When $L^{0}({\cal F},R)$ is endowed with the topology of convergence in…

We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four…

Let $n\ge1$ and $B\ge2$. A real-valued function $f$ defined on the $n$-simplex $\Delta_n$ is approximately convex with respect to $\Delta_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B \in \Delta_n$ and…

Let M be a Seifert manifold which belongs to the geometry Flat. In this work we determine all the free involutions {\tau} on M, and the Borsuk-Ulam indice of (M,{\tau}).

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} +…

We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages of sequences with terms in a given finite set A. For every such set we completely characterize the numbers x ("intermediate values") with…

We study $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hulls of finite sets of points in $\mathbb{R}^3$, as in KirchheimMullerSverak2003. This notion of convexity, which we call $2+1$ convexity, corresponds to rank-one convex convexity,…