13 research outputs found
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
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
Small polygons with large area
A polygon is \textit{small} if it has unit diameter. The maximal area of a
small polygon with a fixed number of sides is not known when is even
and . We determine an improved lower bound for the maximal area of a
small -gon for this case. The improvement affects the term of an
asymptotic expansion; prior advances affected less significant terms. This
bound cannot be improved by more than . For , , , and
, the polygon we construct has maximal area.Comment: 12 page