1,357 research outputs found
On the representation of polyhedra by polynomial inequalities
A beautiful result of Br\"ocker and Scheiderer on the stability index of
basic closed semi-algebraic sets implies, as a very special case, that every
-dimensional polyhedron admits a representation as the set of solutions of
at most polynomial inequalities. Even in this polyhedral case,
however, no constructive proof is known, even if the quadratic upper bound is
replaced by any bound depending only on the dimension.
Here we give, for simple polytopes, an explicit construction of polynomials
describing such a polytope. The number of used polynomials is exponential in
the dimension, but in the 2- and 3-dimensional case we get the expected number
.Comment: 19 pages, 4 figures; revised version with minor changes proposed by
the referee
Polynomial inequalities representing polyhedra
Our main result is that every n-dimensional polytope can be described by at
most (2n-1) polynomial inequalities and, moreover, these polynomials can
explicitly be constructed. For an n-dimensional pointed polyhedral cone we
prove the bound 2n-2 and for arbitrary polyhedra we get a constructible
representation by 2n polynomial inequalities.Comment: 9 page
Densest Lattice Packings of 3-Polytopes
Based on Minkowski's work on critical lattices of 3-dimensional convex bodies
we present an efficient algorithm for computing the density of a densest
lattice packing of an arbitrary 3-polytope. As an application we calculate
densest lattice packings of all regular and Archimedean polytopes.Comment: 37 page
Implications of a wavelength dependent PSF for weak lensing measurements
The convolution of galaxy images by the point-spread function (PSF) is the
dominant source of bias for weak gravitational lensing studies, and an accurate
estimate of the PSF is required to obtain unbiased shape measurements. The PSF
estimate for a galaxy depends on its spectral energy distribution (SED),
because the instrumental PSF is generally a function of the wavelength. In this
paper we explore various approaches to determine the resulting `effective' PSF
using broad-band data. Considering the Euclid mission as a reference, we find
that standard SED template fitting methods result in biases that depend on
source redshift, although this may be remedied if the algorithms can be
optimised for this purpose. Using a machine-learning algorithm we show that, at
least in principle, the required accuracy can be achieved with the current
survey parameters. It is also possible to account for the correlations between
photometric redshift and PSF estimates that arise from the use of the same
photometry. We explore the impact of errors in photometric calibration, errors
in the assumed wavelength dependence of the PSF model and limitations of the
adopted template libraries. Our results indicate that the required accuracy for
Euclid can be achieved using the data that are planned to determine photometric
redshifts
Cone-volume measures of polytopes
The cone-volume measure of a polytope with centroid at the origin is proved
to satisfy the subspace concentration condition. As a consequence a conjectured
(a dozen years ago) fundamental sharp affine isoperimetric inequality for the
U-functional is completely established -- along with its equality conditions.Comment: Slightly revised version thanks to the suggestions of the referees
and other readers; two figures adde
Integer Knapsacks: Average Behavior of the Frobenius Numbers
The main result of the paper shows that the asymptotic growth of the
Frobenius number in average is significantly slower than the growth of the
maximum Frobenius number
- …