In \cite{Sz11} we have generalized the notion of the simplicial density
function for horoballs in the extended hyperbolic space $\bar{\mathbf{H}}^n,
~(n \ge 2)$, where we have allowed {\it congruent horoballs in different types}
centered at the various vertices of a totally asymptotic tetrahedron.

We identify least-perimeter unit-area tilings of the plane by convex
pentagons, namely tilings by Cairo and Prismatic pentagons, find infinitely
many, and prove that they minimize perimeter among tilings by convex polygons
with at most five sides.

The main purpose of the paper is to construct a sequence of graphs of
constant degree with indefinitely growing girths admitting embeddings into
$\ell_1$ with uniformly bounded distortions. This result answers the problem
posed by N. Linial, A. Magen, and A. Naor (2002).

We provide an algorithm for computing the centered Hausdorff measure of
self-similar sets satisfying the strong separation condition. We prove the
convergence of the algorithm and test its utility on some examples.

The basic object of the paper is the moduli space $M_{2,3}(L)$ of a closed
polygonal linkage either in $\mathbb{R}^2$ or in $\mathbb{R}^3$. As was
originally suggested by G. Khimshiashvili, the space $M_{2}(L)$ is equipped
with the oriented area function $A$, whereas (as is suggested in the paper)
$M_{3}(L)$ is equipped with the vector area function $S$. The latter are
generically Morse functions, whose critical points have a nice description.

In this paper, a frame is introduced on tangent bundle of a Finsler manifold
in a manner that it makes some simplicity to study the properties of the
natural foliations in tangent bundle. Moreover, we show that the indicatrix
bundle of a Finsler manifold with lifted sasaki metric and natural almost
complex structure on tangent bundle cannot be a sasakian manifold.

Data sets are often modeled as point clouds in $R^D$, for $D$ large. It is
often assumed that the data has some interesting low-dimensional structure, for
example that of a $d$-dimensional manifold $M$, with $d$ much smaller than $D$.
When $M$ is simply a linear subspace, one may exploit this assumption for
encoding efficiently the data by projecting onto a dictionary of $d$ vectors in
$R^D$ (for example found by SVD), at a cost $(n+D)d$ for $n$ data points.

We prove that each coarsely homogenous separable metric space $X$ is coarsely
equivalent to one of the spaces: the sigleton, the Cantor macro-cube or the
Baire macro-space.

In this paper, we provide a lower bound for an area of the convex hull of
points and a rectangle in a plane. We then apply this estimate to establish a
lower bound for a universal cover problem. We showed that a convex universal
cover for a unit length curve has area at least 0.232239. In addition, we show
that a convex universal cover for a unit closed curve has area at least
0.0879873.

In the packing-constrained point covering problem, PC^2, one seeks
configurations of points in the plane that cannot all be covered by a packing
arrangement of unit disks. We consider in particular the problem of finding the
minimum number of points N for which such a configuration exists and obtain the
bounds 11 <= N <= 55. The disparity of these bounds is symptomatic, we believe,
of the fact that PC^2 belongs in a higher complexity class than the standard
packing and covering problems.

Put n nonoverlapping squares inside the unit square. Let f(n) and g(n) denote
the maximum values of the sum of the edge lengths of the n small squares, where
in the case of f(n) the maximum is taken over all arbitrary packings of the
unit square, and in the case of g(n) it is taken over all tilings of the unit
square (i.e., the total area of the n small squares is 1). Benton and Tyler
asked for which values of n we have f(n)=g(n). We show that f(8)>g(8). More
precisely, we show that g(8)=13/5; it is known that f(8) is at least 8/3.

While the concept of straight-line length is well understood in taxicab
geometry, little research has been done into the length of curves or the nature
of area and volume in this geometry. This paper sets forth a comprehensive view
of the basic dimensional measures in taxicab geometry.

A natural analogue to angles and trigonometry is developed in taxicab
geometry. This structure is then analyzed to see which, if any, congruent
triangle relations hold. A nice application involving the use of parallax to
determine the exact (taxicab) distance to an object is also discussed.

This article provides a simple pictorial introduction to universal hyperbolic
geometry. We explain how to understand the subject using only elementary
projective geometry, augmented by a distinguished circle. This provides a
completely algebraic framework for hyperbolic geometry, valid over the rational
numbers (and indeed any field not of characteristic two), and gives us many new
and beautiful theorems. These results are accurately illustrated with colour
diagrams, and the reader is invited to check them with ruler constructions and
measurements.

Let A be a bounded subset of IR^d. We give an upper bound on the volume of
the symmetric difference of A and f(A) where f is a translation, a rotation, or
the composition of both, a rigid motion. The volume is measured by the
d-dimensional Hausdorff measure, which coincides with the Lebesgue measure for
Lebesgue measurable sets. We bound the volume of the symmetric difference of A
and f(A) in terms of the (d-1)-dimensional volume of the boundary of A and the
maximal distance of a boundary point to its image under f.

In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory
metric and with a Haar measure, we consider isodiametric sets, i.e. sets
maximizing the measure among all sets with a given diameter. In particular,
given an isodiametric set, and up to negligible sets, we prove that its
boundary is given by the graphs of two locally Lipschitz functions. Moreover,
in the restricted class of rotationally invariant sets, we give a quite
complete characterization of any compact (rotationally invariant) isodiametric
set.

A simple proof of Thue theorem on Circle Packing is given. The proof is only
based on density analysis of Delaunay triangulation for the set of points that
are centers of circles in a saturated circle configuration.

A configuration p in r-dimensional Euclidean space is a finite collection of
points (p^1,...,p^n) that affinely span R^r. A bar framework, denoted by G(p),
in R^r is a simple graph G on n vertices together with a configuration p in
R^r.

We present a method for proving upper bounds on the eigenvalues of the graph
Laplacian. A main step involves choosing an appropriate ``Riemannian'' metric
to uniformize the geometry of the graph. In many interesting cases, the
existence of such a metric is shown by examining the combinatorics of special
types of flows. This involves proving new inequalities on the crossing number
of graphs.

We obtain an upper bound to the packing density of regular tetrahedra. The
bound is obtained by showing the existence, in any packing of regular
tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that
each sphere is not fully covered by the packing. The bound on the amount of
space that is not covered in each sphere is obtained in a recursive way by
building on the observation that non-overlapping regular tetrahedra cannot
subtend a solid angle of $4\pi$ around a point if this point lies on a
tetrahedron edge.

Two theorems are presented concerning the Miquel point configuration, when
the operative points on the sides of the triangle are the feet of Cevians,

Let $\H$ denote the discrete Heisenberg group, equipped with a word metric
$d_W$ associated to some finite symmetric generating set. We show that if
$(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function
$f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and
\begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim
\left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation}

Upper bounds for the eigenvalues of the Laplace-Beltrami operator on a
hypersurface bounding a domain in some ambient Riemannian manifold are given in
terms of the isoperimetric ratio of the domain. These results are applied to
the extrinsic geometry of isometric embeddings.

In this paper, we present a simple factor 6 algorithm for approximating the
optimal multiplicative distortion of embedding a graph metric into a tree
metric (thus improving and simplifying the factor 100 and 27 algorithms of
B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi,
Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor
algorithm for approximating the optimal distortion of embedding a graph metric
into an outerplanar metric.

Distance measuring is a very important task in digital geometry and digital
image processing. Due to our natural approach to geometry we think of the set
of points that are equally far from a given point as a Euclidean circle. Using
the classical neighbourhood relations on digital grids, we get circles that
greatly differ from the Euclidean circle. In this paper we examine different
methods of approximating the Euclidean circle in the square grid, considering
the possible motivations as well.

A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks.

In this paper we prove the Kneser-Poulsen conjecture for the case of large
radii. Namely, if a finite number of points in an n-dimensional Euclidean space
is rearranged so that the distance between each pair of points does not
decrease, then there exists a positive number $r_0$ that depends on the
rearrangement of the points, such that if we consider n-dimensional balls of
radius r>r_0 with centers at these points, then the volume of the union
(intersection) of the balls before the rearrangement is not less (no more) than
the volume of the union (intersection) after the rearrangement.

The goal of this paper is to give a purely geometric proof of a theorem by
Branko Gr\"unbaum concerning configuration of triangles coming from the
classical Napoleon's theorem in planar Euclidean geometry.

We introduce some analytic relations on the set of partial differential
equations of two variables. It relies on a new comparison method to give rough
asymptotic estimates for solutions which obey different partial differential
equations. It uses a kind of scale transform called tropical geometry which
connects automata with real rational dynamics. Two different solutions can be
considered when their defining equations are transformed to the same automata
at infinity.

A convex figures F is called uniquely constructed if it satisfies the
following condition: if F equidecomposable to a convex figure G then F is
congruent to G. We classify all convex uniquely constructed figures. The paper
written primary for school students.

We examine topological properties of pointed metric measure spaces $(Y, p)$
that can be realized as the pointed Gromov-Hausdorff limit of a sequence of
complete, Riemannian manifolds $\{(M^n_i, p_i)\}_{i=1}^{\infty}$ with
nonnegative Ricci curvature. Cheeger and Colding \cite{ChCoI} showed that given
such a sequence of Riemannian manifolds it is possible to define a measure
$\nu$ on the limit space $(Y, p)$. In the current work, we generalize previous
results of the author to examine the relationship between the topology of $(Y,
p)$ and the volume growth of $\nu$.

We survey connections between the theory of bi-Lipschitz embeddings and the
Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest
Cut Problem is a striking example of the deep interplay between analysis,
geometry, and probability on the one hand, and computational issues in discrete
mathematics on the other. We explain how the key ideas evolved over the past 20
years, emphasizing the interactions with Banach space theory, geometric measure
theory, and geometric group theory.

We introduce a randomized iterative fragmentation procedure for finite metric
spaces, which is guaranteed to result in a polynomially large subset that is
$D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed
target distortion. Since this procedure works for $D$ arbitrarily close to the
nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much
simpler probabilistic proof of the main result of Bartel, Linial, Mendel, and
Naor, answering a question from Mendel and Naor, and yielding the best known
bounds in the nonlinear Dvoretzky theorem.

We calculate the Assouad dimension of the self-affine carpets of Bedford and
McMullen. We also calculate the conformal Assouad dimension of those carpets
that are not self-similar.

We construct examples of smooth 4-dimensional manifolds M supporting a
locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats
condition, and contain 2-dimensional flats F with the property that the
boundary at infinity of F defines a nontrivial knot in the boundary at infinity
of X. As a consequence, we obtain that the fundamental group of M cannot be
isomorphic to the fundamental group of any Riemannian manifold of nonpositive
sectional curvature.

We introduce smooth L^\infty differential forms on a singular (semialgebraic)
set X in R^n. Roughly speaking, a smooth L^\infty differential form is a
certain class of equivalence of 'stratified forms', that is, a collection of
smooth forms on disjoint smooth subsets (stratification) of X with matching
tangential components on the adjacent strata and bounded size (in the metric
induced from R^n). We identify the singular homology of X as the homology of
the chain complex generated by semialgebraic singular simplices, i.e.
continuous semialgebraic maps from the standard simplices into X.

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to
exactly two edges and every other point belonging to at most one edge. Brass
asked the following question: For $n \geq 5$ odd, what is the maximum perimeter
of a simple $n$-gon contained in a Euclidean unit disk?

A line g is a transversal to a family F of convex polytopes in 3-dimensional
space if it intersects every member of F. If, in addition, g is an isolated
point of the space of line transversals to F, we say that F is a pinning of g.
We show that any minimal pinning of a line by convex polytopes such that no
face of a polytope is coplanar with the line has size at most eight. If, in
addition, the polytopes are disjoint, then it has size at most six. We
completely characterize configurations of disjoint polytopes that form minimal
pinnings of a line.

We consider a natural non-negative two-form G on quasifuchsian space that
extends the Weil-Petersson metric on Teichmuller space. We describe completely
the positive definite locus of G, showing that it is a positive definite metric
off the fuchsian diagonal of quasifuchsian space and is only zero on the
"pure-bending'' tangent vectors to the fuchsian diagonal . We show that G is
equal to the pullback of the pressure metric from dynamics.

The thirteen spheres problem is asking if 13 equal size nonoverlapping
spheres in three dimensions can touch another sphere of the same size. This
problem was the subject of the famous discussion between Isaac Newton and David
Gregory in 1694. The problem was solved by Schutte and van der Waerden only in
1953.

The Fermat-Weber center of a planar body $Q$ is a point in the plane from
which the average distance to the points in $Q$ is minimal. We first show that
for any convex body $Q$ in the plane, the average distance from the
Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot
\Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490
\cdot \Delta(Q)$.

A discrete set in the $p$-dimensional Euclidian space is {\it almost
periodic}, if the measure with the unite masses at points of the set is almost
periodic in the weak sense. We propose to construct positive almost periodic
discrete sets as an almost periodic perturbation of a full rank discrete
lattice. Also we prove that each almost periodic discrete set on the real axes
is an almost periodic perturbation of some arithmetic progression.

In 1994, Martin Gardner stated a set of questions concerning the dissection
of a square or an equilateral triangle in three similar parts. Meanwhile,
Gardner's questions have been generalized and some of them are already solved.
In the present paper, we solve more of his questions and treat them in a much
more general context. Let $D\subset \mathbb{R}^d$ be a given set and let
$f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such
that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a
non-overlapping union?

We give a rigorous computer-assisted proof that the triangular bi-pyramid is
the unique configuration of 5 points on the 2-sphere that globally minimizes
the Coulomb (1/r) potential. We also prove the same result for the (1/r^2)
potential. The main mathematical contribution of the paper is a fairly
efficient energy estimate that works for any number of points and any power-law
potential.

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype
of a Banach space to all metric spaces. Every Banach space has metric cotype at
least 2. We show that any metric space that is bi-Lipschitz equivalent to an
ultrametric space has infinimal metric cotype 1. We discuss the invariance of
metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff
limits, and use these facts to establish a partial converse of the main result.

Our main theorem is an extension of the well-known Mizoguchi-Takahaashi's
fixed point theorem [N. Mizogochi and W. Takahashi, Fixed point theorems for
multi-valued mappings on complete metric space,

{\it J. Math. Anal. Appl.} 141 (1989) 177--188].

The Square Peg Problem asks whether every continuous simple closed planar
curve contains the four vertices of a square. This paper proves this for the
largest so far known class of curves.

For a given symmetrically normed ideal I on an infinite dimensional Hilbert
space H, we study the rectifiable distance in the classical Banach-Lie unitary
group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that
one-parameter subgroups of U_I are short paths, provided the spectrum of the
exponent is bounded by $\pi$, and that any two elements of U_I can be joined
with a short path, thus obtaining a Hopf-Rinow theorem in this infinite
dimensional setting, for a wide and relevant class of (non necessarily smooth)
metrics.

It is proved that the simplex is a strict local minimum for the
volume-product P(K)=min vol(K)vol(K^z), in the Banach-Mazur space of
n-dimensional (classes of) convex bodies. Here K^z is the polar body of K about
the point z and the minimum is taken over all the points z in the interior of
K. Linear local stability in the neighborhood of the simplex is proved as well.
In the proof, methods that were recently introduced by Nazarov, Petrov,
Ryabogin and Zvavitch are extended to the non-symmetric setting.

We study in this work flat surfaces with conical singularities, that is,
surfaces provided with a flat structure with conical singular points. Finding
good parameters for these surfaces in the general case is an open question. We
give an answer to this question in the case of flat structures on pairs of
pants with one singular point. The question of decomposability of an arbitrary
flat surface into flat pairs of pants is discussed.

Let $A_1,A_2,...,A_n$ be the vertices of a polygon with unit perimeter, that
is $\sum_{i=1}^n |A_i A_{i+1}|=1$. We derive various tight estimates on the
minimum and maximum values of the sum of pairwise distances, and respectively
sum of pairwise squared distances among its vertices. Such estimates on these
sums in the literature were known only for convex polygons. We also sharpen a
previous lower bound on the minimum sum of pairwise squared distances for
convex polygons due to Novotn\'y.

This work is devoted to the geometric analysis of metric-measure spaces
satisfying a Prekopa-Leindler or a more general Borell-Brascamp-Lieb
inequality.

Let $X$ be an $s$-distance set in the Euclidean space $\mathbb{R}^d$, and
$A(X)=\{\alpha_1, \alpha_2, ..., \alpha_s \}$ be the set of the Euclidean
distances between two distinct elements of $X$. For $s=2$, Larman-Rogers-Seidel
proved that if $|X| \geq 2 d+4$, then there exists an integer $k$ such that
${\alpha_1}^2/{\alpha_2}^2=(k-1)/k$, that is,
${\alpha_2}^2/({\alpha_2}^2-{\alpha_1}^2)=k$. In this paper, for any $s$, we
give a generalization of the theorem due to Larman-Rogers-Seidel.

It is shown that convex hypersurfaces in Hilbert spaces have nonnegative
Alexandrov curvature. This extends an earlier result of Buyalo for convex
hypersurfaces in Riemannian manifolds of finite dimension.

For a k-flat F inside a locally compact CAT(0)-space X, we identify various
conditions that ensure that F bounds a (k+1)-dimensional half flat in X. Our
conditions are formulated in terms of the ultralimit of X.

We consider the geometry of four spatial displacements, arranged in cyclic
order, such that the relative motion between neighbouring displacements is a
pure rotation. We compute the locus of points whose homologous images lie on a
circle, the locus of oriented planes whose homologous images are tangent to a
cone of revolution, and the locus of oriented lines whose homologous images
form a skew quadrilateral on a hyperboloid of revolution.

We give a direct, pointwise proof for the tube formula of Lapidus-Pearse for
self-similar fractals, where we allow non-convex, non-Steiner-like generators.

In this article we encode Hadwiger's covering conjecture into a series of
functions defined on the spaces of convex bodies, propose a four-step program
to approach this conjecture, and obtain some partial results.

If $X$ is a convex surface in a Euclidean space, then the squared intrinsic
distance function $\dist^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in
the only natural extrinsic sense. An analogous result holds for the squared
distance function $\dist^2(x,F)$ from a closed set $F \subset X$. Applications
concerning $r$-boundaries (distance spheres) and the ambiguous locus
(exoskeleton) of a closed subset of a convex surface are given.

We introduce ideas from geometric group theory related to boundaries of
groups. This is a mostly expository paper. We consider the visual boundary of a
free abelian group, and show that it is an uncountable set with the trivial
topology.

We show that if the joints of a bar and joint framework $(G,p)$ are
positioned as `generically' as possible subject to given symmetry constraints
and $(G,p)$ possesses a `fully-symmetric' infinitesimal flex (i.e., the
velocity vectors of the infinitesimal flex remain unaltered under all symmetry
operations of $(G,p)$), then $(G,p)$ also possesses a finite flex which
preserves the symmetry of $(G,p)$ throughout the path. This and other related
results are obtained by symmetrizing techniques described by L.

A topological space is nonseparably connected if it is connected but all of
its connected separable subspaces are singletons. We show that each connected
first countable space is the image of a nonseparably connected complete metric
space under a continuous monotone hereditarily quotient map.

We establish a generic formula for the generalised q-dimensions of measures
supported by almost self-affine sets, for all q>1. These q-dimensions may
exhibit phase transitions as q varies. We first consider general measures and
then specialise to Bernoulli and Gibbs measures. Our method involves estimating
expectations of moment expressions in terms of `multienergy' integrals which we
then bound using induction on families of trees.

A significant range of geometric structures whose rigidity is explored for
both practical and theoretical purposes are formed by modifying generically
isostatic triangulated spheres. In the block and hole structures (P, p), some
edges are removed to make holes, and other edges are added to create rigid
sub-structures called blocks. Previous work noted a combinatorial analogy in
which blocks and holes played equivalent roles. In this paper, we connect
stresses in such a structure (P, p) to first-order motions in a swapped
structure (P', p), where holes become blocks and blocks become holes.

The notion of the magnitude of a compact metric space was considered in
arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line
segments, circles and Cantor sets. In this paper more evidence is presented for
a conjectured relationship with a geometric measure theoretic valuation.
Firstly, a heuristic is given for deriving this valuation by considering
'large' subspaces of Euclidean space and, secondly, numerical approximations to
the magnitude are calculated for squares, disks, cubes, annuli, tori and
Sierpinski gaskets.

Fractal sets, by definition, are non-differentiable, however their dimension
can be continuous, differentiable, and arithmetically manipulable as function
of their construction parameters. A new arithmetic for fractal dimension of
polyadic Cantor sets is introduced by means of properly defining operators for
the addition, subtraction, multiplication, and division. The new operators have
the usual properties of the corresponding operations with real numbers.

A linear deformation of a Meyer set $M$ in $\RR^d$ is the image of $M$ under
a group homomorphism of the group $[M]$ generated by $M$ into $\RR^d$. We
provide a necessary and sufficient condition for such a deformation to be a
Meyer set. In the case that the deformation is a Meyer set and the deformation
is injective, the deformation is pure point diffractive if the orginal set $M$
is pure point diffractive.

For an arbitrary cocompact hyperbolic Coxeter group G with finite generator
set S and complete growth function P(x)/Q(x), we provide a recursion formula
for the coefficients of the denominator polynomial Q(x) which allows to
determine recursively the Taylor coefficients and the pole behavior of the
growth function of G in terms of its Coxeter subgroup structure. We illustrate
this for compact right-angled hyperbolic n-polytopes.

For $d\geq 2$, we construct a doubling measure $\nu$ on $\R^d$ and a
rectifiable curve $\Gamma$ such that $\nu(\Gamma)>0$.

It is shown that there is no quasi-sphere packing of the lattice grid Z^{d+1}
or a co-compact hyperbolic lattice of H^{d+1} or the 3-regular tree \times Z,
in R^d, for all d. A similar result is proved for some other graphs too. Rather
then using a direct geometrical approach, the main tools we are using are from
non-linear potential theory.

Given a finite regular graph G=(V,E) and a metric space (X,d_X), let
$gamma_+(G,X) denote the smallest constant $\gamma_+>0$ such that for all
f,g:V\to X we have:

\frac{1}{|V|^2}\sum_{x,y\in V} d_X(f(x),g(y))^2\le \frac{\gamma_+}{|E|}
\sum_{xy\in E} d_X(f(x),g(y))^2.

Two tetrahedra are called orthologic if the lines through vertices of one and
perpendicular to corresponding faces of the other are intersecting. This is
equivalent to the orthogonality of non-corresponding edges. We prove that the
additional assumption of intersecting non-corresponding edges (``orthosecting
tetrahedra'') implies that the six intersection points lie on a sphere. To a
given tetrahedron there exists generally a one-parametric family of
orthosecting tetrahedra.

We prove that, for every $k=1,2,...,$ every shortest-path metric on a graph
of pathwidth $k$ embeds into a distribution over random trees with distortion
at most $c$ for some $c=c(k)$. A well-known conjecture of Gupta, Newman,
Rabinovich, and Sinclair states that for every minor-closed family of graphs
$F$, there is a constant $c(F)$ such that the multi-commodity max-flow/min-cut
gap for every flow instance on a graph from $F$ is at most $c(F)$. The
preceding embedding theorem is used to prove this conjecture whenever the
family $F$ does not contain all trees.

Bennett, DeVore and Sharpley introduced the space weak $L^{\infty}$ in 1981
and studied its relationship with functions of bounded mean oscillation. Here
we characterize weak $L^{\infty}$ in measure spaces without using the
decreasing rearrangement of a function. Instead, we obtain exponential
estimates for the distribution function. In addition, we consider a localized
version of the characterization that leads to a new characterization of BMO.

In this paper using the concept of the extended Hamming code we give a
construction for dense packing of points at distance at least one in such unit
cubes which dimension are a power of two.

The concept of separation by hyperplanes is fundamental for convex geometry
and its tropical (max-plus) analogue. However, analogous separation results in
max-min convex geometry are based on semispaces. This paper answers the
question which semispaces are hyperplanes and when it is possible to
classically separate by hyperplanes in max-min convex geometry.

We show that for every $\alpha > 0$, there exist $n$-point metric spaces
(X,d) where every "scale" admits a Euclidean embedding with distortion at most
$\alpha$, but the whole space requires distortion at least $\Omega(\sqrt{\alpha
\log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight,
and disproves a conjecture stated there. This matching upper bound was known to
be tight at both endpoints, i.e. when $\alpha = \Theta(1)$ and $\alpha =
\Theta(\log n)$, but nowhere in between.

Sequences of polygons generated by performing iterative processes on an
initial polygon have been studied extensively. One of the most popular
sequences is the one sometimes referred to as {\it Kasner polygons}. Given a
polygon $K$, the first Kasner descendant $K'$ of $K$ is obtained by placing the
vertices of $K'$ at the midpoints of the edges of $K$.

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of
a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension
and the volume of the submanifold and the order of the eigenvalue, these bounds
depend on either the maximal number of intersection points of $M$ with a
$p$-plane in a generic position (transverse to $M$), or an invariant which
measures the concentration of the volume of $M$ in $\R^{m+p}$. These bounds are
asymptotically optimal in the sense of the Weyl law.

In this paper we complete the proof of the "equivalence" of non-discrete
R-buildings of types A~_2 and C~_2, with, respectively, projective planes and
generalized quadrangles with non-discrete valuation, begun in previous paper of
the authors. We also complete the proof of the "equivalence" of an affine
building of rank 3 with a generalized polygon with discrete valuation (by
proving this for generalized hexagons). We also complement a result of the
second author by proving uniqueness up to scalar multiples of the weight
sequences of polygons with non-discrete valuation.

It is well-known that a paracompact space X is of covering dimension n if and
only if any map f from X to a simplicial complex K can be pushed into its
n-skeleton. We use the same idea to define dimension in the coarse category. It
turns out the analog of maps f from X to K is related to asymptotically
Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's
Property A, and the dimension coincides with Gromov's asymptotic dimension.

Two consequences of the stability version of the one dimensional
Pr\'ekopa-Leindler inequality are presented. One is the stability version of
the Blaschke-Santal\'o inequality, and the other is a stability version of the
Pr\'ekopa-Leindler inequality for even functions in higher dimensions, where a
recent stability version of the Brunn-Minkowski inequality is also used in an
essential way.

We study the problem of acute triangulations of convex polyhedra and the
space R^n. Here an acute triangulation is a triangulation into simplices whose
dihedral angles are acute. We prove that acute triangulations of the n-cube do
not exist for n>=4. Further, we prove that acute triangulations of the space
R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a
construction of an acute triangulation of the cube, the regular octahedron and
a non-trivial acute triangulation of the regular tetrahedron.

We prove two generalizations of results proved by Bruhat and Tits involving
metrical completeness and R-buildings. Firstly, we give a generalization of the
Bruhat-Tits fixed point theorem also valid for non-complete R-buildings, but
with the added condition that the group is finitely generated. Secondly, we
generalize a criterion which reduces the problem of completeness to the wall
trees of the R-building.

We examine the generic local and global rigidity of various graphs in R^d. We
analyze two classes of graphs that satisfy Hendrickson's conditions for generic
global rigidity, yet fail to be generically globally rigid. We find a large
family of bipartite graphs with d > 3, and we use a method that generates
infinitely many graphs in R^5. Finally, we state some conjectures for further
exploration.

We prove that a convex subcomplex of a spherical building of type E7 or E8 is
a subbuilding or the group of building automorphisms preserving the subcomplex
has a fixed point in it. Together with previous results of Muehlherr-Tits, and
Leeb and the author, this completes the proof of Tits' Center Conjecture for
thick spherical buildings.

We study simplices with equiareal faces in the Euclidean 3-space by means of
elementary geometry. We present an unexpectedly simple proof of the fact that,
if such a simplex is non-degenerate, than every two of its faces are congruent.
We show also that this statement is wrong for degenerate simplices and find all
degenerate simplices with equiareal faces.

We prove that Alexandrov spaces $X$ of nonnegative curvature have Markov type
2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a
subset of $X$ into a 2-uniformly convex Banach space is extended as a Lipschitz
continuous map on the entire space $X$.

Hyperbolic geometry is developed in a purely algebraic fashion from first
principles, without a prior development of differential geometry. The natural
connection with the geometry of Lorentz, Einstein and Minkowski comes from a
projective point of view, with trigonometric laws that extend to `points at
infinity', here called `null points', and beyond to `ideal points' associated
to a hyperboloid of one sheet. The theory works over a general field not of
characteristic two, and the main laws can be viewed as deformations of those
from planar rational trigonometry.

The paper contains a characterization of compact groups $G\subseteq\GL(V)$,
where $V$ is a finite dimensional real vector space, which have the following
property \SP{}: the family of convex hulls of $G$-orbits is a semigroup with
respect to the Minkowski addition. If $G$ is finite, then \SP{} holds if and
only if $G$ is a Coxeter group; if $G$ is connected then \SP{} is true if and
only if $G$ is polar. In general, $G$ satisfies \SP{} if and only if it is
polar and its Weyl group is a Coxeter group.

In this paper we introduce a family of examples that can be regarded as
spaces of nonpositive curvature, but with the distinct quality that they are
not complete as metric spaces. This amounts to the fact that they are modelled
on a finite von Neumann algebra, and the metrics introduced arise from the
trace of the algebra. In spite of the noncompleteness of these manifolds, their
geometry can be studied from the view-point of metric geometry, and several
techniques derived from the functional analysis are applied to gain insight on
their geodesic structure.

We introduce the wedge product of two polytopes. The wedge product is
described in terms of inequality systems, in terms of vertex coordinates as
well as purely combinatorially, from the corresponding data of its
constituents. The wedge product construction can be described as an iterated
``subdirect product'' as introduced by McMullen (1976); it is dual to the
``wreath product'' construction of Joswig and Lutz (2005).

The similar sublattices of a planar lattice can be classified via its
multiplier ring. The latter is the ring of rational integers in the generic
case, and an order in an imaginary quadratic field otherwise. Several classes
of examples are discussed, with special emphasis on concrete results. In
particular, we derive Dirichlet series generating functions for the number of
distinct similar sublattices of a given index, and relate them to various zeta
functions of orders in imaginary quadratic fields.