3,420 research outputs found
Transversal numbers over subsets of linear spaces
Let be a subset of . It is an important question in the
theory of linear inequalities to estimate the minimal number such that
every system of linear inequalities which is infeasible over has a
subsystem of at most inequalities which is already infeasible over
This number is said to be the Helly number of In view of Helly's
theorem, and, by the theorem due to Doignon, Bell and
Scarf, We give a common extension of these equalities
showing that We show that
the fractional Helly number of the space (with the
convexity structure induced by ) is at most as long as
is finite. Finally we give estimates for the Radon number of mixed
integer spaces
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points in a
projective plane determines at least lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points in a projective space
determines at least hyperplanes, unless all the points are contained in a
hyperplane. Let be a spanning subset of a -dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of ,
there are at least as many -dimensional subspaces as there are
-dimensional subspaces, for every at most . This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
-adic intersection complexes.Comment: 18 pages, major revisio
Quantitative Tverberg theorems over lattices and other discrete sets
This paper presents a new variation of Tverberg's theorem. Given a discrete
set of , we study the number of points of needed to guarantee the
existence of an -partition of the points such that the intersection of the
convex hulls of the parts contains at least points of . The proofs
of the main results require new quantitative versions of Helly's and
Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:1503.0611
Free nilpotent and -type Lie algebras. Combinatorial and orthogonal designs
The aim of our paper is to construct pseudo -type algebras from the
covering free nilpotent two-step Lie algebra as the quotient algebra by an
ideal. We propose an explicit algorithm of construction of such an ideal by
making use of a non-degenerate scalar product. Moreover, as a bypass result, we
recover the existence of a rational structure on pseudo -type algebras,
which implies the existence of lattices on the corresponding pseudo -type
Lie groups. Our approach substantially uses combinatorics and reveals the
interplay of pseudo -type algebras with combinatorial and orthogonal
designs. One of the key tools is the family of Hurwitz-Radon orthogonal
matrices
Quantitative Tverberg, Helly, & Carath\'eodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg,
Helly, & Caratheodory theorems in combinatorial convexity. Our results include
measurable or enumerable information in the hypothesis and the conclusion.
Typical measurements include the volume, the diameter, or the number of points
in a lattice.Comment: 33 page
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