11 research outputs found
Problems in Convex Geometry
We deal with five different problems from convex geometry, each on its own chapter of this Thesis. These problems are the following. Random copies of a convex body: We study the probability that a random copy of a convex body intersects the integer lattice in a certain way. A conjecture by Erdos: We study the statement by Erdos "On every convex curve there exists a point P such that every circle with centre P intersects the curve in at most 2 points." A Yao-Yao type theorem: Given a nice measure in R^d, we show that there is a partition P of R^d into 3*2^(d/2) convex pieces of equal measure such that every hyperplane avoids at least 2 elements of P. Line transversals: Given a family F of balls in R^d such that every three have a transversal line, we bound the blow-up factor l needed so that lF has a line transversal. Longest lattice convex chains: Given a triangle with two specified vertices v_1, v_2 in Z^2, we bound the size of the largest lattice convex chain from v_1 to v_2. The techniques used to tackle these problems are very diverse and include results from analysis, combinatorics, number theory and topology, as well as the use of computers
Helly numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset
. We contribute new results about their -Helly
numbers. We extend prior work for , , and ; we give sharp bounds on the -Helly numbers in
several new cases. We considered the situation for low-dimensional and for
sets that have some algebraic structure, in particular when is an
arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz
method we obtain colorful versions of many monochromatic Helly-type results,
including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was
originally the first half of arXiv:1504.00076v
Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of -optimization
The scenario approach developed by Calafiore and Campi to attack
chance-constrained convex programs utilizes random sampling on the uncertainty
parameter to substitute the original problem with a representative continuous
convex optimization with convex constraints which is a relaxation of the
original. Calafiore and Campi provided an explicit estimate on the size of
the sampling relaxation to yield high-likelihood feasible solutions of the
chance-constrained problem. They measured the probability of the original
constraints to be violated by the random optimal solution from the relaxation
of size .
This paper has two main contributions. First, we present a generalization of
the Calafiore-Campi results to both integer and mixed-integer variables. In
fact, we demonstrate that their sampling estimates work naturally for variables
restricted to some subset of . The key elements are
generalizations of Helly's theorem where the convex sets are required to
intersect . The size of samples in both algorithms will
be directly determined by the -Helly numbers.
Motivated by the first half of the paper, for any subset , we introduce the notion of an -optimization problem, where the
variables take on values over . It generalizes continuous, integer, and
mixed-integer optimization. We illustrate with examples the expressive power of
-optimization to capture sophisticated combinatorial optimization problems
with difficult modular constraints. We reinforce the evidence that
-optimization is "the right concept" by showing that the well-known
randomized sampling algorithm of K. Clarkson for low-dimensional convex
optimization problems can be extended to work with variables taking values over
.Comment: 16 pages, 0 figures. This paper has been revised and split into two
parts. This version is the second part of the original paper. The first part
of the original paper is arXiv:1508.02380 (the original article contained 24
pages, 3 figures
Erdős–Szekeres Theorem for Lines
According to the Erdős–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erdős–Szekeres theorem. © 2015, Springer Science+Business Media New York
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Helly numbers of algebraic subsets of Rd and an extension of doignon's theorem
We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in Rd with a proper subset S ? Rd, and contribute new results about their S-Helly numbers. We extend prior work for S = Rd, Zd, and Zd-k × Rk, and give some sharp bounds for several new cases: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of Rd or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-Type results, including several colorful versions of our own results
Helly numbers of algebraic subsets of ℝ d
We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in Rd with a proper subset S ? Rd, and contribute new results about their S-Helly numbers. We extend prior work for S = Rd, Zd, and Zd-k × Rk, and give some sharp bounds for several new cases: low-dimensional situations, sets that have some algebraic structure, in particular when S is an arbitrary subgroup of Rd or when S is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lovász method we obtain colorful versions of many monochromatic Helly-Type results, including several colorful versions of our own results
Recommended from our members
Helly numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset . We contribute new results about their -Helly numbers. We extend prior work
for , , and ; we give sharp
bounds on the -Helly numbers in several new cases. We considered the situation for
low-dimensional and for sets that have some algebraic structure, in particular when
is an arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we
obtain colorful versions of many monochromatic Helly-type results, including several
colorful versions of our own results
A rainbow Ramsey analogue of Rado's theorem
We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use new techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature
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Beyond Chance-Constrained Convex Mixed-Integer Optimization: A Generalized Calafiore-Campi Algorithm and the notion of -optimization
The scenario approach developed by Calafiore and Campi to attack chance-constrained
convex programs utilizes random sampling on the uncertainty parameter to substitute the
original problem with a representative continuous convex optimization with convex
constraints which is a relaxation of the original. Calafiore and Campi provided an explicit
estimate on the size of the sampling relaxation to yield high-likelihood feasible
solutions of the chance-constrained problem. They measured the probability of the original
constraints to be violated by the random optimal solution from the relaxation of size .
This paper has two main contributions. First, we present a generalization of the
Calafiore-Campi results to both integer and mixed-integer variables. In fact, we
demonstrate that their sampling estimates work naturally for variables restricted to some
subset of . The key elements are generalizations of Helly's theorem where
the convex sets are required to intersect . The size of samples in
both algorithms will be directly determined by the -Helly numbers. Motivated by the
first half of the paper, for any subset , we introduce the notion of
an -optimization problem, where the variables take on values over . It generalizes
continuous, integer, and mixed-integer optimization. We illustrate with examples the
expressive power of -optimization to capture sophisticated combinatorial optimization
problems with difficult modular constraints. We reinforce the evidence that
-optimization is "the right concept" by showing that the well-known randomized sampling
algorithm of K. Clarkson for low-dimensional convex optimization problems can be extended
to work with variables taking values over