20,043 research outputs found
The Alexandrov-Fenchel type inequalities, revisited
Various Alexandrov-Fenchel type inequalities have appeared and played
important roles in convex geometry, matrix theory and complex algebraic
geometry. It has been noticed for some time that they share some striking
analogies and have intimate relationships. The purpose of this article is to
shed new light on this by comparatively investigating them in several aspects.
\emph{The principal result} in this article is a complete solution to the
equality characterization problem of various Alexandrov-Fenchel type
inequalities for intersection numbers of nef and big classes on compact
K\"{a}hler manifolds, extending earlier results of Boucksom-Favre-Jonsson,
Fu-Xiao and Xiao-Lehmann. Our proof combines a result of Dinh-Nguy\^{e}n on
K\"{a}hler geometry and an idea in convex geometry tracing back to Shephard. In
addition to this central result, we also give a geometric proof of the complex
version of the Alexandrov-Fenchel type inequality for mixed discriminants and a
determinantal type generalization of various Alexandrov-Fenchel type
inequalities.Comment: 18 pages, slightly revised version stressing our principal result,
comments welcom
Good covers are algorithmically unrecognizable
A good cover in R^d is a collection of open contractible sets in R^d such
that the intersection of any subcollection is either contractible or empty.
Motivated by an analogy with convex sets, intersection patterns of good covers
were studied intensively. Our main result is that intersection patterns of good
covers are algorithmically unrecognizable.
More precisely, the intersection pattern of a good cover can be stored in a
simplicial complex called nerve which records which subfamilies of the good
cover intersect. A simplicial complex is topologically d-representable if it is
isomorphic to the nerve of a good cover in R^d. We prove that it is
algorithmically undecidable whether a given simplicial complex is topologically
d-representable for any fixed d \geq 5. The result remains also valid if we
replace good covers with acyclic covers or with covers by open d-balls.
As an auxiliary result we prove that if a simplicial complex is PL embeddable
into R^d, then it is topologically d-representable. We also supply this result
with showing that if a "sufficiently fine" subdivision of a k-dimensional
complex is d-representable and k \leq (2d-3)/3, then the complex is PL
embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in
version
Multiplicative combinatorial properties of return time sets in minimal dynamical systems
We investigate the relationship between the dynamical properties of minimal
topological dynamical systems and the multiplicative combinatorial properties
of return time sets arising from those systems. In particular, we prove that
for a residual sets of points in any minimal system, the set of return times to
any non-empty, open set contains arbitrarily long geometric progressions. Under
the separate assumptions of total minimality and distality, we prove that
return time sets have positive multiplicative upper Banach density along
and along multiplicative subsemigroups of ,
respectively. The primary motivation for this work is the long-standing open
question of whether or not syndetic subsets of the positive integers contain
arbitrarily long geometric progressions; our main result is some evidence for
an affirmative answer to this question.Comment: 32 page
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
Dual Mixed Volumes and the Slicing Problem
We develop a technique using dual mixed-volumes to study the isotropic
constants of some classes of spaces. In particular, we recover, strengthen and
generalize results of Ball and Junge concerning the isotropic constants of
subspaces and quotients of L_p and related spaces. An extension of these
results to negative values of p is also obtained, using generalized
intersection-bodies. In particular, we show that the isotropic constant of a
convex body which is contained in an intersection-body is bounded (up to a
constant) by the ratio between the latter's mean-radius and the former's
volume-radius. We also show how type or cotype 2 may be used to easily prove
inequalities on any isotropic measure.Comment: 38 pages, to appear in Advances in Mathematics. Corrected Remark 4.
Topological transversals to a family of convex sets
Let be a family of compact convex sets in . We say
that has a \emph{topological -transversal of index }
(, ) if there are, homologically, as many transversal
-planes to as -planes containing a fixed -plane in
.
Clearly, if has a -transversal plane, then
has a topological -transversal of index for and . The converse is not true in general.
We prove that for a family of compact convex sets in
a topological -transversal of index implies an
ordinary -transversal. We use this result, together with the
multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann
category of the Grassmannian, and different versions of the colorful Helly
theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences
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