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1. CMB 2013 (vol 57 pp. 640)
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. 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 |
3. 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 |
4. 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 |
5. 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 |
6. CMB 2002 (vol 45 pp. 634)
Local Complexity of Delone Sets and Crystallinity This paper characterizes when a Delone set $X$ in $\mathbb{R}^n$ is an
ideal crystal in terms of restrictions on the number of its local
patches of a given size or on the heterogeneity of their distribution.
For a Delone set $X$, let $N_X (T)$ count the number of
translation-inequivalent patches of radius $T$ in $X$ and let
$M_X(T)$ be the minimum radius such that every closed ball of radius
$M_X(T)$ contains the center of a patch of every one of these kinds.
We show that for each of these functions there is a
``gap in the spectrum'' of possible growth rates between being
bounded and having linear growth, and that having sufficiently
slow linear growth is equivalent to $X$ being an ideal crystal.
Explicitly, for $N_X(T)$, if $R$ is the covering radius of $X$
then either $N_X(T)$ is bounded or $N_X (T) \ge T/2R$ for all $T>0$.
The constant $1/2R$ in this bound is best possible in all dimensions.
For $M_X(T)$, either $M_X(T)$ is bounded or $M_X(T)\ge T/3$ for all $T>0$.
Examples show that the constant $1/3$ in this bound cannot be replaced by
any number exceeding $1/2$. We also show that every aperiodic Delone
set $X$ has $M_X(T)\ge c(n) T$ for all $T>0$, for a certain constant $c(n)$
which depends on the dimension $n$ of $X$ and is $>1/3$ when $n>1$.
Keywords:aperiodic set, Delone set, packing-covering constant, sphere packing Categories:52C23, 52C17 |