5,594 research outputs found
Metric inequalities for polygons
Let be the vertices of a polygon with unit perimeter, that
is . 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 -gon ( 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
Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
In this paper we develop an integer-affine classification of
three-dimensional multistory completely empty convex marked pyramids. We apply
it to obtain the complete lists of compact two-dimensional faces of
multidimensional continued fractions lying in planes with integer distances to
the origin equal 2, 3, 4 ... The faces are considered up to the action of the
group of integer-linear transformations. In conclusion we formulate some actual
unsolved problems associated with the generalizations for n-dimensional faces
and more complicated face configurations.Comment: Minor change
The number of unit distances is almost linear for most norms
We prove that there exists a norm in the plane under which no n-point set
determines more than O(n log n log log n) unit distances. Actually, most norms
have this property, in the sense that their complement is a meager set in the
metric space of all norms (with the metric given by the Hausdorff distance of
the unit balls)
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