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## Abstract

Let *S*
^{
d-1} denote the (*d* − 1)-dimensional unit sphere centered at the origin of the *d*-dimensional Euclidean space. Let 0 < α < π. A set P of points in *S*
^{
d-1} is called almost α-equidistant if among any three points of P there is at least one pair lying at spherical distance α. In
this note we prove upper bounds on the cardinality of P depending only on *d*.

## Abstract

*c*<

*s*be fixed real numbers such that

*f*: E

^{2}→ E

^{ d }for

*d*≥ 2 be a function such that for every

*p*,

*q*∈

**E**

^{2}if

*p − q*=

*c*, then

*f*(

*p*) −

*f*(

*q*) ≤

*c*, and if

*p − q*=

*s*, then

*f*(

*p*) −

*f*(

*q*) ≥

*s*. Then

*f*is a congruence. This result depends on and expands a result of Rdo et. al. [9], where a similar result holds, but for

^{2}is replaced by E

^{ n }for

*n*> 2 and where the range of

*c/s*is enlarged.

## Abstract

In this paper we prove some stronger versions of Danzer-Grnbaum's theorem including the following stability-type result.
For 0 < α < 14π/27 the maximum number of vertices of a convex polyhedron in **E**
^{3} such that all angles between adjacent edges are bounded from above by α is 8. One of the main tools is the spherical geometry
version of Pl's theorem.

## Abstract

The main aim of this paper is to prove that the maximal operator *σ*
_{0}
^{
k*}:= sup_{
n
} ∣*σ*
_{
n,n
}
^{
k
}∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space *H*
_{1/2} to the space weak-*L*
_{1/2}.

## Abstract

*N*) and q = (q1, q2, …, q

*N*) are two configurations in

^{ d }(p

_{ i },

*r*

_{ i }) and B

^{ d }(q

_{ i },

*r*

_{ i }) of radius

*r*

_{ i }, for

*i*= 1, …,

*N*. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For

*d*= 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over

## Abstract

A convex *d*-polytope in ℝ^{
d
} is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes
of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.

## Abstract

The integrals of maximal Riesz and Nörlund kernels are infinite, so we have to use some weight function to “pull them back” to the finite. In this paper we give necessary and sufficient conditions for the weight function to get a finite integral on bounded Vilenkin groups. For our motivation we refer the readers to [4], [5], [6].