11 research outputs found
On the size of lattice simplices with a single interior lattice point
Let be the set of all -dimensional simplices in
with integer vertices and a single integer point in the interior of
. It follows from a result of Hensley that is finite up
to affine transformations that preserve . It is known that, when
grows, the maximum volume of the simplices T \in \cT^d(1) becomes
extremely large. We improve and refine bounds on the size of (where by the size we mean the volume or the number of
lattice points). It is shown that each can be
decomposed into an ascending chain of faces whose sizes are `not too large'.
More precisely, if , then there exist faces of such that, for every ,
is -dimensional and the size of is bounded from above in terms
of and . The bound on the size of is double exponential in .
The presented upper bounds are asymptotically tight on the log-log scale.Comment: accepted in SIAM J. Discrete Mat
Lattice point inequalities for centered convex bodies
We study upper bounds on the number of lattice points for convex bodies
having their centroid at the origin. For the family of simplices as well as in
the planar case we obtain best possible results. For arbitrary convex bodies we
provide an upper bound, which extends the centrally symmetric case and which,
in particular, shows that the centroid assumption is indeed much more
restrictive than an assumption on the number of interior lattice points even
for the class of lattice polytopes.Comment: 12 page
Local optimality of Zaks-Perles-Wills simplices
In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex
S_{d,k} with k interior lattice points, whose volume is linear in k and doubly
exponential in the dimension d. It is conjectured that, for all d \ge 3 and k
\ge 1, the simplex S_{d,k} is a volume maximizer in the family P^d(k) of all
d-dimensional lattice polytopes with k interior lattice points. To obtain a
partial confirmation of this conjecture, one can try to verify it for a
subfamily of P^d(k) that naturally contains S_{d,k} as one of the members.
Currently, one does not even know whether S_{d,k} is optimal within the family
S^d(k) of all d-dimensional lattice simplices with k interior lattice points.
In view of this, it makes sense to look at even narrower families, for example,
some subfamilies of S^d(k). The simplex S_{d,k} of Zaks, Perles and Wills has a
facet with only one lattice point in the relative interior. We show that
S_{d,k} is a volume maximizer in the family of simplices S \in S^d(k) that have
a facet with one lattice point in its relative interior. We also show that, in
the above family, the volume maximizer is unique up to unimodular
transformations
Lattice simplices with a fixed positive number of interior lattice points: A nearly optimal volume bound
We give an explicit upper bound on the volume of lattice simplices with fixed
positive number of interior lattice points. The bound differs from the
conjectural sharp upper bound only by a linear factor in the dimension. This
improves significantly upon the previously best results by Pikhurko from 2001
Classifying terminal weighted projective space
We present a classification of all weighted projective spaces with at worst
terminal or canonical singularities in dimension four. As a corollary we also
classify all four-dimensional one-point lattice simplices up to equivalence.
Finally, we classify the terminal Gorenstein weighted projective spaces up to
dimension ten.Comment: 19 pages, 6 table
Optimizing Sparsity over Lattices and Semigroups
Motivated by problems in optimization we study the sparsity of the solutions
to systems of linear Diophantine equations and linear integer programs, i.e.,
the number of non-zero entries of a solution, which is often referred to as the
-norm. Our main results are improved bounds on the -norm of
sparse solutions to systems , where ,
and is either a general integer vector (lattice case)
or a non-negative integer vector (semigroup case). In the lattice case and
certain scenarios of the semigroup case, we give polynomial time algorithms for
computing solutions with -norm satisfying the obtained bounds
Optimizing sparsity over lattices and semigroups
Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the ℓ0-norm. Our main results are improved bounds on the ℓ0-norm of sparse solutions to systems Ax=b, where A∈Zm×n, b∈Zm and x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In the lattice case and certain scenarios of the semigroup case, we give polynomial time algorithms for computing solutions with ℓ0-norm satisfying the obtained bounds
On finite generation and infinite convergence of generalized closures from the theory of cutting planes
For convex sets and in we define to be the
convex hull of all points belonging to but not to the interior of .
Cutting-plane methods from integer and mixed-integer optimization can be
expressed in geometric terms using functionals with appropriately chosen
sets . We describe the geometric properties of and characterize
those for which maps polyhedra to polyhedra. For certain natural
classes of convex sets in we consider the
functional given by . The functional can be used to define
various types of closure operations considered in the theory of cutting planes
(such as the Chv\'atal closure, the split closure as well as generalized split
closures recently introduced by Andersen, Louveaux and Weismantel). We study
conditions on under which maps rational
polyhedra to rational polyhedra. We also describe the limit of the sequence of
sets obtained by iterative application of to . A part of
the presented material gives generalized formulations and unified proofs of
several recent results obtained by various authors
Amoebas of genus at most one
The amoeba of a Laurent polynomial f \in \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}]
is the image of its zero set under the log-absolute-value map.
Understanding the space of amoebas (i.e., the decomposition of the space of all
polynomials, say, with given support or Newton polytope, with regard to the
existing complement components) is a widely open problem.
In this paper we investigate the class of polynomials whose Newton
polytope \New(f) is a simplex and whose support contains exactly one
point in the interior of \New(f). Amoebas of polynomials in this class may
have at most one bounded complement component. We provide various results on
the space of these amoebas. In particular, we give upper and lower bounds in
terms of the coefficients of for the existence of this complement component
and show that the upper bound becomes sharp under some extremal condition. We
establish connections from our bounds to Purbhoo's lopsidedness criterion and
to the theory of -discriminants.
Finally, we provide a complete classification of the space of amoebas for the
case that the exponent of the inner monomial is the barycenter of the simplex
Newton polytope. In particular, we show that the set of all polynomials with
amoebas of genus 1 is path-connected in the corresponding space of amoebas,
which proves a special case of the question on connectivity (for general Newton
polytopes) stated by H. Rullg{\aa}rd.Comment: 26 pages, 5 figures; minor revisio
Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results
For each dimension , -dimensional integral simplices with exactly one
interior integral point have bounded volume. This was first shown by Hensley.
Explicit volume bounds were determined by Hensley, Lagarias and Ziegler,
Pikhurko, and Averkov. In this paper we determine the exact upper volume bound
for such simplices and characterize the volume-maximizing simplices. We also
determine the sharp upper bound on the coefficient of asymmetry of an integral
polytope with a single interior integral point. This result confirms a
conjecture of Hensley from 1983. Moreover, for an integral simplex with
precisely one interior integral point, we give bounds on the volumes of its
faces, the barycentric coordinates of the interior integral point and its
number of integral points. Furthermore, we prove a bound on the lattice
diameter of integral polytopes with a fixed number of interior integral points.
The presented results have applications in toric geometry and in integer
optimization.Comment: Advances in Mathematic