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1. CMB Online first
Equilateral sets and a SchÃ¼tte Theorem for the $4$-norm A well-known theorem of SchÃ¼tte (1963) gives a sharp lower bound for
the ratio of the maximum and minimum distances between $n+2$ points in
$n$-dimensional Euclidean space.
In this note we adapt BÃ¡rÃ¡ny's elegant proof (1994) of this theorem to the space $\ell_4^n$.
This gives a new proof that the largest cardinality of an equilateral
set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an
interval $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$.
Categories:46B20, 52A21, 52C17 |
2. CMB 2012 (vol 57 pp. 61)
2-dimensional Convexity Numbers and $P_4$-free Graphs For $S\subseteq\mathbb R^n$ a set
$C\subseteq S$ is an $m$-clique if the convex hull of no $m$-element subset of
$C$ is contained in $S$.
We show that there is essentially just one way to construct
a closed set $S\subseteq\mathbb R^2$ without an uncountable
$3$-clique that is not the union of countably many convex sets.
In particular, all such sets have the same convexity number;
that is, they
require the same number of convex subsets to cover them.
The main result follows from an analysis of the convex structure of closed
sets in $\mathbb R^2$ without uncountable 3-cliques in terms of
clopen, $P_4$-free graphs on Polish spaces.
Keywords:convex cover, convexity number, continuous coloring, perfect graph, cograph Categories:52A10, 03E17, 03E75 |
3. CMB 2012 (vol 57 pp. 178)
Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.
Keywords:density topology, quasiconvex function, approximate continuity, point of continuity Categories:52A41, 26B05 |
4. CMB 2012 (vol 57 pp. 42)
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.
Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces Categories:46B20, 46C05, 52C17 |
5. CMB 2012 (vol 57 pp. 3)
A Short Proof of Paouris' Inequality We give a short proof of a result of G.~Paouris on
the tail behaviour of the Euclidean norm $|X|$ of an isotropic
log-concave random vector $X\in\mathbb{R}^n,$
stating that for every $t\geq 1$,
\[\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).\]
More precisely we show that for any log-concave random vector $X$
and any $p\geq 1$,
\[(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in
S^{n-1}}(\mathbb{E} |\langle
z,X\rangle|^p)^{1/p}.\]
Keywords:log-concave random vectors, deviation inequalities Categories:46B06, 46B09, 52A23 |
6. CMB 2011 (vol 55 pp. 498)
Simplices in the Euclidean Ball We establish some inequalities for the second moment
$$
\frac{1}{|K|} \int_{K}|x|_2^2 \,dx
$$
of a convex body $K$ under various assumptions on the position of $K$.
Keywords:convex body, simplex Category:52A20 |
7. CMB 2011 (vol 55 pp. 697)
Constructions of Uniformly Convex Functions We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.
Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm Categories:52A41, 46G05, 46N10, 49J50, 90C25 |
8. CMB 2011 (vol 55 pp. 767)
On Zindler Curves in Normed Planes We extend the notion of Zindler curve from the Euclidean plane to
normed planes. A characterization of Zindler curves for general
normed planes is given, and the relation between Zindler curves and
curves of constant area-halving distances in such planes is
discussed.
Keywords:rc length, area-halving distance, Birkhoff orthogonality, convex curve, halving pair, halving distance, isosceles orthogonality, midpoint curve, Minkowski plane, normed plane, Zindler curve Categories:52A21, 52A10, 46C15 |
9. CMB 2011 (vol 55 pp. 487)
Weighted Model Sets and their Higher Point-Correlations Examples of distinct weighted model sets with equal $2,3,4, 5$-point
correlations are given.
Keywords:model sets, correlations, diffraction Categories:52C23, 51P05, 74E15, 60G55 |
10. CMB 2011 (vol 55 pp. 98)
Similarity and Coincidence Isometries for Modules The groups of (linear) similarity and coincidence isometries of
certain modules $\varGamma$ in $d$-dimensional Euclidean space, which
naturally occur in quasicrystallography, are considered. It is shown
that the structure of the factor group of similarity modulo
coincidence isometries is the direct sum of cyclic groups of prime
power orders that divide $d$. In particular, if the dimension $d$ is a
prime number $p$, the factor group is an elementary abelian
$p$-group. This generalizes previous results obtained for lattices to
situations relevant in quasicrystallography.
Categories:20H15, 82D25, 52C23 |
11. CMB 2011 (vol 54 pp. 726)
Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a
normed linear space $Y$. A symmetric, bounded, closed, convex set
$A$ in a finite dimensional normed linear space $X$ is called a
sufficient enlargement for $X$ if, for an arbitrary
isometric embedding of $X$ into a Banach space $Y$, there exists a
linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each
finite dimensional normed space has a minimal-volume sufficient
enlargement that is a parallelepiped; some spaces have ``exotic''
minimal-volume sufficient enlargements. The main result of the
paper is a characterization of spaces having ``exotic''
minimal-volume sufficient enlargements in terms of Auerbach
bases.
Keywords:Banach space, Auerbach basis, sufficient enlargement Categories:46B07, 52A21, 46B15 |
12. CMB 2010 (vol 54 pp. 561)
A Note on Toric Varieties Associated with Moduli Spaces
In this note we give a brief review of the construction of a toric
variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface
$\Sigma^g$ equipped with a trinion, or pair of pants, decomposition.
This was outlined by J. Hurtubise and L.~C. Jeffrey.
A. Tyurin used this construction on a certain
collection of trinion decomposed surfaces to produce a variety
$DM_g$, the so-called \emph{Delzant model of moduli space}, for
each genus $g.$ We conclude this note with some basic facts about
the moment polytopes of the varieties $\mathcal{V}.$ In particular,
we show that the varieties $DM_g$ constructed by Tyurin, and claimed
to be smooth, are in fact singular for $g \geq 3.$
Categories:14M25, 52B20 |
13. CMB 2010 (vol 53 pp. 614)
The Mean Width of Circumscribed Random Polytopes
For a given convex body $K$ in ${\mathbb R}^d$, a random polytope
$K^{(n)}$ is defined (essentially) as the intersection of $n$
independent closed halfspaces containing $K$ and having an isotropic
and (in a specified sense) uniform distribution. We prove upper and
lower bounds of optimal orders for the difference of the mean widths
of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial
polytope $P$, a precise asymptotic formula for the difference of the
mean widths of $P^{(n)}$ and $P$ is obtained.
Keywords:random polytope, mean width, approximation Categories:52A22, 60D05, 52A27 |
14. CMB 2010 (vol 53 pp. 394)
On Nearly Equilateral Simplices and Nearly l_{â} Spaces By $\textrm{d}(X,Y)$ we denote the (multiplicative) Banach--Mazur distance between two normed spaces $X$ and $Y.$ Let $X$ be an $n$-dimensional normed space with $\textrm{d}(X,\ell_\infty^n) \le 2,$ where $\ell_\infty^n$ stands for $\mathbb{R}^n$ endowed with the norm $\|(x_1,\dots,x_n)\|_\infty := \max \{|x_1|,\dots, |x_n| \}.$ Then every metric space $(S,\rho)$ of cardinality $n+1$ with norm $\rho$ satisfying the condition $\max D / \min D \le 2/ \textrm{d}(X,\ell_\infty^n)$ for $D:=\{ \rho(a,b) : a, b \in S, \ a \ne b\}$ can be isometrically embedded into $X.$
Categories:52A21, 51F99, 52C99 |
15. CMB 2009 (vol 53 pp. 3)
A Combinatorial Reciprocity Theorem for Hyperplane Arrangements Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of
linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in
$\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha
\in \mathcal A$
and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that
the coefficients of the characteristic polynomial of $\mathcal A^m$ are
quasi-polynomials in $m$ and that they satisfy a simple combinatorial
reciprocity law.
Categories:52C35, 05E99 |
16. CMB 2009 (vol 52 pp. 403)
Shaken Rogers's Theorem for Homothetic Sections We shall prove the following shaken Rogers's theorem for
homothetic sections: Let $K$ and $L$ be strictly convex bodies and
suppose that for every plane $H$ through the origin we can choose
continuously sections of $K $ and $L$, parallel to $H$, which are
directly homothetic. Then $K$ and $L$ are directly homothetic.
Keywords:convex bodies, homothetic bodies, sections and projections, Rogers's Theorem Category:52A15 |
17. CMB 2009 (vol 52 pp. 327)
Geometric ``Floral'' Configurations With an increase in size, configurations of points and lines
in the plane usually become complicated and hard to analyze.
The ``floral'' configurations we are introducing here represent
a new type that makes accessible and visually intelligible
even configurations of considerable size. This is achieved
by combining a large degree of symmetry with a hierarchical
construction. Depending on the details of the interdependence
of these aspects, there are several subtypes that are described
and investigated.
Categories:52C30, 52C99 |
18. CMB 2009 (vol 52 pp. 342)
On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width The X-ray numbers of some classes of convex bodies are investigated.
In particular, we give a proof of the X-ray Conjecture as well as
of the Illumination Conjecture for almost smooth convex bodies
of any dimension and for convex bodies of constant width of
dimensions $3$, $4$, $5$ and $6$.
Keywords:almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray Conjecture Categories:52A20, 52A37, 52C17, 52C35 |
19. CMB 2009 (vol 52 pp. 349)
On Projection Bodies of Order One The projection body of order one $\Pi_1K$ of a convex body $K$ in
$\R^n$ is the body whose support function is, up to a constant, the
average mean width of the orthogonal projections of $K$ onto
hyperplanes through the origin.
The paper contains an inequality for the support function of
$\Pi_1K$, which implies in particular that such a function is
strictly convex, unless $K$ has dimension one or two. Furthermore,
an existence problem related to the reconstruction of a convex body
is discussed to highlight the different behavior of the area
measures of order one and of order $n-1$.
Category:52A40 |
20. CMB 2009 (vol 52 pp. 361)
A Note on Covering by Convex Bodies A classical theorem of Rogers states
that for any convex body $K$ in $n$-dimensional Euclidean space
there exists a covering of the space by translates of $K$ with
density not exceeding $n\log{n}+n\log\log{n}+5n$. Rogers' theorem
does not say anything about the structure of such a covering. We
show that for sufficiently large values of $n$ the same bound can
be attained by a covering which is the union of $O(\log{n})$
translates of a lattice arrangement of $K$.
Categories:52C07, 52C17 |
21. CMB 2009 (vol 52 pp. 366)
A Class of Cellulated Spheres with Non-Polytopal Symmetries We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$.
We prove that these cellulations cannot be polytopal with maximal
combinatorial symmetry. Such non-realizability phenomenon was first
described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and,
to the knowledge of the author, until now there have not been any
known examples in higher dimensions. As a starting point for the
construction, we introduce a new class of (Wythoffian) uniform
polytopes, which we call duplexes. In proving our main result,
we use some tools that we developed earlier while studying perfect
polytopes. In particular, we prove perfectness of the duplexes;
furthermore, we prove and make use of the perfectness of another
new class of polytopes which we obtain by a variant of the so-called
$E$-construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope Categories:52B11, 52B15, 52B70 |
22. CMB 2009 (vol 52 pp. 380)
Successive Minima and Radii In this note we present inequalities relating the successive minima of an
$o$-symmetric convex body and the successive inner and outer radii of the
body. These inequalities join known inequalities involving only either
the successive minima or the successive radii.
Keywords:successive minima, inner and outer radii Categories:52A20, 52C07, 52A40, 52A39 |
23. CMB 2009 (vol 52 pp. 407)
On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies The Bezdek--Pach conjecture asserts that the maximum number of
pairwise touching positive homothetic copies of a convex body in
$\Re^d$ is $2^d$. Nasz\'odi proved that the quantity in question is
not larger than $2^{d+1}$. We present an improvement to this result by
proving the upper bound $3\cdot2^{d-1}$ for centrally symmetric
bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a
convex body. We extend this definition, prove an upper bound on the
resulting quantity, and show a connection with the problem of touching
homothetic bodies.
Keywords:Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodality Categories:52C17, 51N20, 51K05, 52A21, 52A37 |
24. CMB 2009 (vol 52 pp. 424)
Covering Discs in Minkowski Planes We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of side-lengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$-segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 |
25. CMB 2009 (vol 52 pp. 451)
Indecomposable Coverings We prove that for every $k>1$, there exist $k$-fold coverings of the
plane (i) with strips, (ii) with axis-parallel rectangles, and
(iii) with homothets of any fixed concave quadrilateral, that cannot
be decomposed into two coverings. We also construct for every
$k>1$ a set of points $P$ and a family of disks $\cal D$ in the
plane, each containing at least $k$ elements of $P$, such that, no
matter how we color the points of $P$ with two colors,
there
exists a disk $D\in{\cal D}$ all of whose points are of the same
color.
Categories:52C15, 05C15 |