301 research outputs found

### Metric inequalities for polygons

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. In most cases such
estimates on these sums in the literature were known only for convex polygons.
In the second part, we turn to a problem of Bra\ss\ regarding the maximum
perimeter of a simple $n$-gon ($n$ odd) contained in a disk of unit radius. The
problem was solved by Audet et al. \cite{AHM09b}, who gave an exact formula.
Here we present an alternative simpler proof of this formula. We then examine
what happens if the simplicity condition is dropped, and obtain an exact
formula for the maximum perimeter in this case as well.Comment: 13 pages, 2 figures. This version replaces the previous version from
8 Feb 2011. A new section has been added and the material has been
reorganized; a correction has been done in the proof of Lemma 4 (analysis of
Case 3

### Approximate Euclidean Ramsey theorems

According to a classical result of Szemer\'{e}di, every dense subset of
$1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large
enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson
says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large
grid, if $N$ is large enough. Here we generalize these results for separated
point sets on the line and respectively in the Euclidean space: (i) every dense
separated set of points in some interval $[0,L]$ on the line contains an
arbitrary long approximate arithmetic progression, if $L$ is large enough. (ii)
every dense separated set of points in the $d$-dimensional cube $[0,L]^d$ in
\RR^d contains an arbitrary large approximate grid, if $L$ is large enough. A
further generalization for any finite pattern in \RR^d is also established.
The separation condition is shown to be necessary for such results to hold. In
the end we show that every sufficiently large point set in \RR^d contains an
arbitrarily large subset of almost collinear points. No separation condition is
needed in this case.Comment: 11 pages, 1 figure

### The opaque square

The problem of finding small sets that block every line passing through a
unit square was first considered by Mazurkiewicz in 1916. We call such a set
{\em opaque} or a {\em barrier} for the square. The shortest known barrier has
length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower
bound for the length of a (not necessarily connected) barrier is $2$, as
established by Jones about 50 years ago. No better lower bound is known even if
the barrier is restricted to lie in the square or in its close vicinity. Under
a suitable locality assumption, we replace this lower bound by $2+10^{-12}$,
which represents the first, albeit small, step in a long time toward finding
the length of the shortest barrier. A sharper bound is obtained for interior
barriers: the length of any interior barrier for the unit square is at least $2
+ 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established
by Sylvester for the measure of all lines that meet two disjoint planar convex
bodies, and (ii) a procedure for detecting lines that are witness to the
invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure

### Sweeping an oval to a vanishing point

Given a convex region in the plane, and a sweep-line as a tool, what is best
way to reduce the region to a single point by a sequence of sweeps? The problem
of sweeping points by orthogonal sweeps was first studied in [2]. Here we
consider the following \emph{slanted} variant of sweeping recently introduced
in [1]: In a single sweep, the sweep-line is placed at a start position
somewhere in the plane, then moved continuously according to a sweep vector
$\vec v$ (not necessarily orthogonal to the sweep-line) to another parallel end
position, and then lifted from the plane. The cost of a sequence of sweeps is
the sum of the lengths of the sweep vectors. The (optimal) sweeping cost of a
region is the infimum of the costs over all finite sweeping sequences for that
region. An optimal sweeping sequence for a region is one with a minimum total
cost, if it exists. Another parameter of interest is the number of sweeps.
We show that there exist convex regions for which the optimal sweeping cost
cannot be attained by two sweeps. This disproves a conjecture of Bousany,
Karker, O'Rourke, and Sparaco stating that two sweeps (with vectors along the
two adjacent sides of a minimum-perimeter enclosing parallelogram) always
suffice [1]. Moreover, we conjecture that for some convex regions, no finite
sweeping sequence is optimal. On the other hand, we show that both the 2-sweep
algorithm based on minimum-perimeter enclosing rectangle and the 2-sweep
algorithm based on minimum-perimeter enclosing parallelogram achieve a $4/\pi
\approx 1.27$ approximation in this sweeping model.Comment: 9 pages, 4 figure

- β¦