Talk Titles and Abstracts
Random matrices with independent log-concave rows
by
Radoslaw Adamczak
University of Warsaw / Fields Institute
Coauthors: O. Guedon, A. Litvak, A. Pajor, N. Tomczak-Jaegermann
I will discuss several properties of random matrices with independent
log-concave rows, obtained during the last several years, including
estimates on their norms, a solution to the Kannan-Lovasz-Simonovits
problem and estimates on their smallest singular value. If time
permits I will also briefly mention some limiting results for their
spectral distributions.
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Volume of Lp-zonotopes and best best constants in Brascamp-Lieb
inequalities
by
David Alonso
Universidad de Zaragoza, Fields Institute
Given some unit vectors a1, ..., am ? Rn that span all Rn and some
positive numbers q1, ..., qm, we consider for every p = 1 the convex
body
Kp:={ x ? Rn : ?i=1m |<x, [(ai)/(qi)]>|p = 1}.
We will give some upper bounds for the volume of Kp and some lower
bounds for the volume of its polar, depending on some parameters,
which improve the ones obtained using the Brascamp-Lieb inequality.
We will also see how the best choice of this parameters is related
to the transformation which takes Kp to a special position which,
for instance, when p=8, is John's position.
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Gaussian and almost Gaussian formulas for volumes and the number
of integer points in polytopes
by
Alexander Barvinok
University of Michigan
Coauthors: John A. Hartigan (Yale)
We present a family of computationally efficient formulas for volumes
and the number of integer points in polytopes represented as the
intersection of the non-negative orthant and an affine subspace.
Although the formulas are not always applicable, they are asymptotically
exact in a wide variety of situations. In particular, we obtain
asymptotic formulas for the number of non-negative integer matrices
with prescribed row and column sums and for the volumes of the respective
transportation polytopes. The intuition for the formulas is provided
by the maximum entropy principle, the Local Central Limit Theorem
and its ramifications.
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Invariant distributions in integral geometry
by
Gautier Berck
When the group, or the isotropy subgroup, is non-compact, classical
Crofton type formulae may fail to exist because of the appearence
of divergent integrals. The aim of the talk is to show that in some
situations this problem may be circumvented replacing the invariant
measure by an invariant distribution. The procedure will be illustrated
by basic examples and applications in convex geometry.
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On the location of roots of Steiner polynomials
by
Maria A. Hernandez Cifre
Departamento de Matematicas, Universidad de Murcia
Coauthors: Martin Henk
For two convex bodies K, E of the n-dimensional euclidean space
and a non-negative real number x, the volume of K+xE is a polynomial
of degree n in x, whose coefficients are, up to a constant, important
measures associated to both sets, the relative quermassintegrals.
This polynomial is called the (relative) Steiner polynomial of K
(with respect to E). If we consider the Steiner polynomial as a
formal polynomial in a complex variable, we are interested in studying
geometric properties of its roots: their location in the complex
plane, size, relation with other geometric magnitudes (in- and circumradius)
and characterization of (families of) convex bodies by mean of properties
of the roots. In this talk I will show the known results on this
topic, which had its starting point in a problem posed by Teissier
in 1982, in the context of Algebraic Geometry.
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Integral functionals verifying a Brunn-Minkowski type inequality
by
Andrea Colesanti
Universita' di Firenze
Coauthors: Daniel Hug, Eugenia Saorin-Gomez
We consider a class of integral functionals defined in the family
of convex bodies. The value of the functional on a convex body is
given by the integral of a fixed continuous function defined on
the unit sphere, with respect to the area measure of the convex
body. We assume that a functional of this form verifies an inequality
of Brunn-Minkowski type. We prove that if in addition the functional
is symmetric, then it must be a mixed volume. The same result holds
if the function defining the functional has some regularity property.
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Small ball probability estimates, ?_2-behavior and the hyperplane
conjecture
by
Nikos Dafnis
University of Athens
We introduce a method which leads to upper bounds for the isotropic
constant. We prove that a positive answer to the hyperplane conjecture
is equivalent to some very strong small
probability estimates for the Euclidean norm on isotropic convex
bodies. As a consequence of our method, we obtain an alternative
proof of the result of J. Bourgain that every ?2-body has bounded
isotropic constant, with a slightly better estimate.
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The Poisson summation formula uniquely characterizes the Fourier
Transform
by
Dmitry Faifman
Ph.D student, Tel-Aviv university
We show that under some regularity assumptions, The Poisson summation
formula uniquely defines the Fourier Transform of a function. Then,
we show how a family of unitary operators on L^2[0,infinity) can
be constructed which exhibit Poisson-like summation formulas As
a by-product of this construction, peculiar unitary operators given
by series arise.
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Concentration of measure phenomenon and eigenvalues of Laplacian
by
Kei Funano
Kumamoto university
Coauthors: Takashi Shioya (Tohoku university)
In this talk, we discuss the relation between the concentration
of measure phenomenon and (behavior of) eigenvalues of Laplacian
on a closed Riemannian manifold. M. Gromov and V. D. Milman was
first studied for the case of the first non-trivial eigenvalue of
Laplacian. Under non-negative Ricci curvature assumption we study
the case of the k-th eigenvalues of Laplacian for any k.
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Embedding from lpn into lrN for 0 < r < p < 2.
by
Olivier GUEDON
Université Paris-Est Marne-La-Vallée
Coauthors: Omer FRIEDLAND
We will present a Kashin type result for embedding lpn into l1N
for 1 < p < 2 and N arbitrarily close to n. We will show that
it is possible to define random embedings such that the conclusion
holds with overwhelming probability. The result can also be extended
to embedding from lpn to lrN with 0 < r = 1. One of the main
tool that we developp is a new type of multi-dimensional Esseen
inequality for studying small ball probabilities.
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Minkowski valuations intertwining the special linear group
by
Christoph Haberl
Vienna University of Technology
A classification of SL(n) co- or contravariant Minkowski valuations
will be presented. Here, a Minkowski valuation is a mapping from
convex bodies to convex bodies which satisfies the inclusion-exclusion
principle. Thereby, we obtain new characterizations of the projection
and centroid body operator. Our result shows that the additional
assumption of homogeneity in previous classifications is not necessary.
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The average Frobenius number
by
Martin Henk
University of Magdeburg
Coauthors: Iskander Aliev, Aicke Hinrichs
Given a primitive positive integer vector a ? Zn > 0, the largest
integer that cannot be represented as a non-negative integer combination
of the coefficients of a is called the Frobenius number of a. In
a series of papers V.I. Arnold initiated the research to study the
average size of Frobenius numbers, and in a recent paper, Bourgain
and Sinai showed that the probability of a "large" Frobenius
number is "comparable small". Based on an approach using
methods from Geometry of Numbers we can strengthen this result in
such a way that we can estimate the average size of Frobenius numbers.
Together with a discrete version of a reverse arithmetic-geometric-mean
inequality by Gluskin and Milman, this allows us to show that for
large instances the order of magnitude of the expected Frobenius
number is (up to a constant depending only on the dimension) given
by its lower bound, which, in particular, strengthens a recent result
of Marklof on the asymptotic distribution of Frobenius numbers.
Furthermore, we discuss generalizations to the case of more than
one input vector a.
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Volume and mixed volume inequalities in stochastic geometry
by
Daniel Hug
Karlsruhe Institute of Technology
Coauthors: Karoly Böröczky, Rolf Schneider, ...
Stochastic geometry deals with random structures such as random
closed sets, random processes of flats or random tessellations.
A useful method for analyzing such structures is to associate a
deterministic convex set (sometimes a zonoid) with it. Thus strong
results from convex geometric analysis become available. As appetizers,
we give two examples:
Let Z0 denote the zero cell of a stationary Poisson hyperplane
tessellation. We are interested in sharp bounds for the expected
number of vertices of Z0. Such bounds are provided by the Blaschke-Santaló
inequality and by the Mahler inequality for zonoids. Equality cases
in these bounds characterize special direction distributions of
the given hyperplane tessellation. Recently, these bounds have been
improved by corresponding stability estimates, first in the geometric
and then in the probabilistic framework. (joint work with Károly
Böröczky)
As a second, new example, let X denote a stationary Poisson hyperplane
process with fixed intensity g in \Rn. From X we pass to the intersection
process X(k) of order k, which is a stationary process of (n-k)-flats
in \Rn. It is well known that the intersection density g(k)(X),
i.e. the intensity of X(k), is maximal if and only if X is isotropic.
Here we introduce a measure for the strength of intersections from
an affine-invariant point of view. The problem of determining its
minimal value leads to a novel geometric inequality for mixed volumes
of zonoids with isotropic generating measures. The solution of a
related problem involving joint intersections from a process of
lines and an independent process of hyperplanes is partly based
on Keith Ball's reverse isoperimetric inequality together with the
equality conditions due to Franck Barthe. (joint work with Rolf
Schneider)
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On the extremal distance between two convex bodies
by
C Hugo Jimenez
University of Seville
Coauthors: Márton Naszódi Eötvös University,
Budapest
We consider d(K, L) a modified version of the Banach-Mazur distance
of convex bodies in Rn proposed by Grünbaum. Gordon, Litvak,
Meyer and Pajor in 2004 showed that for any two convex bodies d(K,
L) = n, moreover, if K is a simplex and L=-L then d(K, L)=n. The
following question arises naturally: Is equality only attained when
one of the sets is a simplex? Leichtweiss, and later Palmon proved
that if d(K, B2n)=n, where B2n is the Euclidean ball, then K is
the simplex. We prove the affirmative answer to the question in
the case when one of the bodies is strictly convex or smooth, thus
obtaining a generalization of the result of Leichtweiss and Palmon.
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If you can hide behind it, can you hide inside it?
by
Dan Klain
University of Massachusetts Lowell
Suppose that K and L are compact convex subsets of Euclidean space,
and suppose that, for every direction u, the orthogonal projection
(that is, the shadow) of L onto the subspace u? normal to u contains
a translate of the corresponding projection of the body K. Does
this imply that the original body L contains a translate of K? Can
we even conclude that K has smaller volume than L?
In other words, suppose K can "hide behind" L from any
point of view (and without rotating). Does this imply that K can
"hide inside" the body L? Or, if not, do we know, at least,
that K has smaller volume?
Although these questions have easily demonstrated negative answers
in dimension 2 (since the projections are 1-dimensional, and convex
1-dimensional sets have very little structure), the (possibly surprising)
answer to these questions continues to be No in Euclidean space
of any finite dimension.
In this talk I will give concrete constructions for convex sets
K and L in n-dimensional Euclidean space such that each (n-1)-dimensional
shadow of L contains a translate of the corresponding shadow of
K, while at the same time K has strictly greater volume than L.
This construction turns out to be sufficiently straightforward that
a talented person could conceivably mold 3-dimensional examples
out of modeling clay.
The talk will then address a number of related questions, such
as: under what additional conditions on K or L does shadow covering
imply actual covering? What bounds can be placed on the volume ratio
of K and L if the shadows of L cover those K?
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The chain rule as a functional equation
by
Hermann Koenig
University of Kiel, Germany
Coauthors: S. Artstein-Avidan, V. Milman
Let T be an operator from C^1(R) into C(R) satisfying the chain
rule functional equation T ( f o g) = (T f) o g * (T g) . We show
that in a non-degenerate case any solution of this equation has
the form (T f)(x) = H (f(x)) / H(x)* |f'(x)|^p * {sgn(f'(x)) , where
H is continuous and p > 0 and where the last term sgn(f'(x))
may be missing; then also p = 0 is possible. An "initial condition"
like T(2*Id) = 2 will imply that T f = f' holds. We also consider
T operating on smoother functions C^k(R) or C^inf(R) and n-dimensional
generalizations of the chain rule equation.
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Moments of unconditional logarithmically concave vectors.
by
Rafal Latala
University of Warsaw
Let X=(X_1, X_2, .., X_n) be a random vector with unconditional
logaritmically concave distribution. We will discuss several results
and open problems related to moments of linear combinations of X_i's.
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Rational Ehrhart quasi-polynomials
by
Eva Linke
Otto-von-Guericke-Universität Magdeburg
Ehrhart's famous theorem states that the number of integral points
in a rationalpolytope is a quasi-polynomial in the integral dilation
factor. We present the generalization to rational dilation factors.
The number of integral points can still be written as a rational
quasi-polynomial. Furthermore, the coefficients of this rational
quasi-polynomial are piecewise polynomial functions and related
to each other by derivation.
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On the Euclidean metric entropy.
by
A. Litvak
University of Alberta
Coauthors: V. Milman, A. Pajor, N. Tomczak-Jaegermann
We discuss some properties of entropy and covering numbers. In
particular we show extension and lifting properties. We provide
applications as well.
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Another observation about operator compressions
by
Elizabeth Meckes
Case Western Reserve University
Coauthors: Mark Meckes
Let T be a self-adjoint operator on a finite-dimensional Hilbert
space. It is shown that the distribution of the eigenvalues of a
compression of T to a subspace of a given dimension is almost the
same for almost all subspaces. This is a coordinate-free analogue
of a recent result of Chatterjee and Ledoux on principal submatrices.
The proof is based on measure concentration and entropy techniques,
and the result improves on the result of Chatterjee and Ledoux in
various ways. This is joint work with Mark Meckes.
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Functional inequalities related to Mahler's conjecture
by
Mathieu Meyer
Université de Paris Est Marne-la-Vallée
We develop topics related to the title.
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Properties of isoperimetric, spectral-gap and log-Sobolev inequalities
via concentration
by
Emanuel Milman
University of Toronto
Various properties of isoperimetric, Sobolev-type and concentration
inequalities are studied on a Riemannian manifold equipped with
a measure, whose generalized Ricci curvature is (possibly negatively)
bounded from below.
First, stability of these inequalities with respect to perturbation
of the measure is obtained. The extent of the perturbation is measured
using several different distances between perturbed and original
measure, such as a one-sided L8 bound on the ratio between their
densities, Total-Variation, Wasserstein distances, and relative
entropy (Kullback-Leibler divergence). In particular, an extension
of the Holley-Stroock perturbation lemma for the log-Sobolev inequality
is obtained, and the dependence on the perturbation parameter is
improved from linear to logarithmic.
Next, in the compact setting, an optimal (up to numeric constants)
isoperimetric inequality is obtained as a function of the curvature
lower bound and diameter upper bound. In particular, the best known
log-Sobolev inequality is obtained in this setting.
Time permitting, we will also mention the equivalence of Transport-Entropy
inequalities with different cost functions and some of their applications.
The main tool used is a previous precise result on the equivalence
between concentration and isoperimetric inequalities in the described
setting.
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Hormander's proof of the Bourgain-Milman theorem
by
Fedor Nazarov
University of Wisconsin at Madison
A long standing Mahler's conjecture asserts that the product of
the volumes of a symmetric convex body in Rn and its polar body
is never less than Pn=4n/n!. Bourgain and Milman proved the lower
bound cn Vn with some small positive constant c. Later, Kuperberg
showed that one can take c=p/4. We shall use Hormander's ideas to
give a fairly simple complex-analytic proof of the Bourgain-Milman
theorem.
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Feige's inequality
by
Krzysztof Oleszkiewicz
Institute of Mathematics: University of Warsaw & Polish Academy
of Sciences
Let S denote a sum of (finitely many) independent non-negative
random variables with means not exceeding 1. A remarkable result
of Uriel Feige (SIAM Journal on Computing, 2006) states that for
every 0<t<1 the inequality P(S<ES+t) > Ct holds true,
where C is some universal positive constant (i.e. it does not depend
on t, distributions and number of the random variables).
A short and simple proof will be presented, to a large extent along
the lines of He, Zhang and Zhang (Mathematics of Operation Research,
2010), as well as some generalizations of the result.
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Properties of metric spaces which are not coarsely embeddable
into a Hilbert space
by
Mikhail Ostrovskii
St. Johns University, Queens, NY
The talk is devoted to expansion properties of locally finite metric
spaces which do not embed coarsely into a Hilbert space. The obtained
characterization can be used, for example, to derive the fact that
infinite locally finite graphs excluding a minor embed coarsely
into a Hilbert space.
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On the existence of subgaussian directions for log-concave measures
by
Grigoris Paouris
Texas A&M University
Coauthors: A. Giannopoulos and P. Valettas
We show that if m is a centered log-concave probability measure
on Rn then, [(c1)/(vn)] = |Y2(m)|1/n = c2[(v{logn})/(vn)], where
Y2(m) is the y2-body of m, and c1, c2 > 0 are absolute constants.
It follows that m has "almost subgaussian" directions:
there exists q ? Sn-1 such that m({ x ? Rn : |<x, q>| = c
t E |<·, q>| } ) = e- [( t2)/(log(t+1))] for all 1
= t = v{nlogn}, where c > 0 is an absolute constant.
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On the reconstruction of inscribable sets in discrete tomography
by
Carla Peri
Università Cattolica - Piacenza
Coauthors: Paolo Dulio
In the usual continuous model for tomography one attempts to reconstruct
a function from a knowledge of its line integrals. All the reconstruction
methods used in computerized tomography require a large number of
projection images to obtain results of acceptable quality.
The field of discrete tomography focuses on the reconstruction
of samples, that consist of only a few different materials, from
a "small" number of projections. For instance, it can
be applied to the reconstruction of nanocrystals at atomic resolution,
where it is assumed that the crystal contains only a few types of
atoms, and that the atoms lie on a regular grid, modeled by the
integer lattice. The high energies required to produce the discrete
X-rays of a crystal mean that just a few number of X-rays can be
taken before the crystal is damaged, so that the conventional techniques
of computerized tomography fail.
In general, this reconstruction task is a ill-posed inverse problem.
In fact, for general data there need not exist a solution, if the
data is consistent, the solution need not be uniquely determined
and "small" changes in the data can lead to unique but
disjoint solutions. Thus, one has to use a priori information, such
as convexity or connectedness, about the sets that have to be reconstructed
to satisfy existence, uniqueness and stability requirements.
By now there are many uniqueness results available for different
classes of finite lattice sets, but just few stability results.
In the present paper we introduce some new classes of lattice sets,
and investigate the problem of their reconstruction by means of
their X-rays taken in the directions belonging to given finite set
D. The geometric structure of such sets enable us to prove results
concerning additivity and uniqueness. When D is the set of coordinate
directions, we give a sharp stability estimate which depends only
on the data error, differently from all the known results, which
also involve the sizes of the sets. Some of these results hold true
in any dimension.
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On the volume of random convex sets
by
Peter Pivovarov
Fields Institute
Coauthors: Grigoris Paouris
Let K ? Rn be a convex body of volume one. Let X1, ..., XN be independent
random vectors distributed uniformly in K and let KN be their (symmetric)
convex hull. A result of Groemer's states that the expected volume
of KN is smallest when K is the Euclidean ball of volume one. A
similar result, due to Bourgain, Meyer, Milman and Pajor, holds
for the volume of random zonotopes ZN=?i=1N Xi. If T:RN?Rn is the
(random) linear operator defined by Tei=Xi, for i=1, ..., N, then
KN is the image of the unit ball in l1N, while ZN is the image of
the unit ball in l8N. What happens when T is applied to other sets?
I will discuss a unified approach to various inequalities involving
the volume of random convex sets for which the Euclidean ball is
the minimizer.
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Poisson-Voronoi approximation
by
Matthias Reitzner
Univ. Osnabrueck
Coauthors: Matthias Schulte
Let X be a Poisson point process and K a convex set. For a point
x in X denote by v(x) the Voronoi cell with respect to X, and by
vX (K) the union of all Voronoi cells with center in K. We call
vX(K) the Poisson-Voronoi approximation of K.
For K a compact convex set the volume difference Vd(vX(K))-Vd(K)
and the symmetric difference Vd(vX(K) \triangle K) are investigated.
Estimates for the variance and central limit theorems are obtained
using the chaotic decomposition of these functions in multiple Wiener-Ito
integrals. (Work in progress jointly with Matthias Schulte)
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Spectral properties of random conjunction matrices
by Mark Rudelson
University of Missouri
Coauthors: Shiva Kasiviswanathan, Adam Smith, and Jon Ullman
We consider a problem in random matrix theory, which arises from
computer science. The standard way to release the statistical summary
of the information contained in a large data base is to publish
its contingency table, which contains percentages of records having
several given common entries. However, if the contingency table
is released exactly, one can reconstruct the individual entries
by solving a system of equations. The standard way to protect the
privacy of individual records is to add a random noise to the contingency
table. Determining the minimal necessary amount of such noise leads
to the problem of estimating the smallest singular value of a special
random matrix with dependent entries, which is generated from a
random matrix with i.i.d. entries taking values 0 and 1.
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Translation Invariant Valuations
by
Franz Schuster
Vienna University of Technology
Coauthors: Semyon Alesker and Andreas Bernig
As a generalization of the notion of measure, valuations on convex
bodies have long played a central role in geometry. The starting
point for many important new results in valuation theory is Hadwiger's
remarkable characterization of the continuous rigid motion invariant
real valued valuations as linear combinations of the intrinsic volumes.
Among many applications, this result allows an effortless proof
of the famous Principal Kinematic Formula from integral geometry.
In the first part of this talk, the decomposition of the space
of continuous translation invariant valuations into a sum of SO(n)
irreducible subspaces is presented. It will be explained how this
result can be reformulated in terms of a Hadwiger type theorem for
translation invariant and SO(n) equivariant valuations with values
in an arbitrary (finite dimensional) SO(n) module. From this perspective
the classical theorem of Hadwiger becomes the special case when
the SO(n) module is the trivial 1-dimensional one.
A striking recent development in valuation theory explores the
connections between isoperimetric inequalities and convex body valued
valuations. To be more specific, many powerful geometric inequalities
involve fundamental operators on convex bodies which are valuations,
e.g. projection and intersection body maps. In many instances the
proofs of these inequalities are based on the symmetry of certain
bivaluations associated with convex body valued valuations.
In the second part, the decomposition of the space of translation
invariant valuations into irreducible SO(n) modules is used to study
the symmetry of O(n) invariant bivaluations and to establish new
Brunn-Minkowski type inequalities for convex body valued valuations.
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Affine Differential Invariants for Convex Bodies
by
Alina Stancu
Concordia University, Montreal
Coauthors: n/a
In what concerns the affine component of Felix Klein’s Erlangen
program, the first results were due to Blaschke’s school. They
were of differential geometric nature and required at least C^4
regularity of closed convex hypersurfaces with, often, positive
Gauss curvature everywhere. While this is unsatisfactory for the
general study of convex bodies, certain objects – the most
famous one being the affine surface area – have appeared in
affine differential geometry but they were later extended to arbitrary
convex bodies with suprising applications. We want to motivate a
certain direction of research which seeks new affine differential
invariants and their applications in view of possible extensions
to arbitrary convex bodies. The talk will not require any prerequisite.
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Estimation of covariance matrices
by
Roman Vershynin
University of Michigan
Estimation of the covariance matrix of a p-dimensional probability
distribution is a basic problem in statistics. A classical estimator
is the sample covariance matrix, obtained from a sample of n independent
points. The more classical regime and well studied regime is where
n > p. We conjecture that n = O(p) suffices to accurately estimate
the covariance matrix of arbitrary distribution with finite 4-th
moments. We discuss some recent progress on this problem, which
has a connection to the "Levy flight", a heavy-tailed
Brownian motion that exhibits sporadic huge jumps (similar to a
predator's path looking for prey). The other regime, n < p, has
recently become quite popular in statistics and its various applications
(e.g. genomics) because of limiting sampling capacities compared
with huge dimensions. We will discuss the problem and recent progress
in this regime as well.
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GL(n) intertwining Minkowski valuations
by
Thomas Wannerer
Vienna University of Technology
Coauthors: Franz E. Schuster
Recently, M. Ludwig characterized continuous Minkowski valuations
which intertwine the general linear group under the assumption that
the valuations either are defined on the set of convex bodies containing
the origin or are invariant under translations. We extend these
results without any restrictions on the domain or invariance under
translations. Part of this work is joint with Franz Schuster.
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Valuations and local functionals
by
Wolfgang Weil
Karlsruhe Institute of Technology
For the classical motion invariant valuations on convex bodies,
the intrinsic volumes, local variants exist, namely measure-valued
additive and motion covariant functionals. These are the curvature
measures. In view of applications in integral geometry, we study
translation invariant functionals on convex bodies or convex polytopes
which admit such a local version. We clarify the connection between
these local functionals and valuations and also discuss whether
valuations on convex polytopes which have certain continuouity properties
allow an extension to all convex bodies.
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Volume Integral Means of Holomorphic Mappings
by
Jie Xiao
Memorial University
Coauthors: Kehe Zhu
In this talk we discuss some geometric aspects of several complex
variables via considering the integral means of holomorphic functions
in the unit complex ball with respect to weighted volume measures,
including Sobolev-type embedding and isoperimetric inequalities
associated to holomorphic maps, log-convexity of the integral means,
and weighted Ricci curvatures.
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Towards an Orlicz Brunn-Minkowski theory
by
Deane Yang
Polytechnic Institute of NYU
Coauthors: Erwin Lutwak and Gaoyong Zhang
Recent extensions of the classical and L_p Brunn-Minkowski theory
to an Orlicz Brunn-Minkowski theory are discussed.
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The geometry of p-convex intersection bodies.
by
Vladyslav Yaskin
University of Alberta
Coauthors: J.Kim and A.Zvavitch
Busemann's theorem states that the intersection body of an origin-symmetric
convex body is also convex. We provide a version of Busemann's theorem
for p-convex bodies. We show that the intersection body of a p-convex
body is q-convex for certain q. Furthermore, we discuss the sharpness
of the previous result by constructing an appropriate example. Finally,
we extend these theorems to some general measure spaces with log-concave
and s-concave measures.
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Shadow boundaries and the Fourier transform
by
Maryna Yaskina
University of Alberta
We investigate the Fourier transform of homogeneous functions on
$\mathbb R^n$ which are not necessarily even. These techniques are
applied to the study of nonsymmetric convex bodies, in particular
to the question of reconstructing convex bodies from the information
about their shadow boundaries.
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On the homothety conjecture
by
Deping Ye
The Fields Institute
Coauthors: E. Werner
Let K be a convex body in Rn and Kd its floating body. The homothety
conjecture asks: "Does Kd=c K imply that K is an ellipsoid?"
Here c is a constant depending on d only. We prove that the homothety
conjecture holds true in the class of the convex bodies Bnp, 1 =
p = 8, the unit balls of lpn; namely, we show that (Bnp)d = c Bnp
if and only if p=2. We also show that the homothety conjecture is
true for a general convex body K if d is small enough.
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Some geometric properties of Intersection Body Operator.
by
Artem Zvavitch
Kent State University
Coauthors: Fedor Nazarov, Dmitry Ryabogin
The notion of an intersection body of a star body was introduced
by E. Lutwak: K is called the intersection body of L if the radial
function of K in every direction is equal to the (d-1)-dimensional
volume of the central hyperplane section of L perpendicular to this
direction.
The notion turned out to be quite interesting and useful in Convex
Geometry and Geometric tomography. It is easy to see that the intersection
body of a ball is again a ball. E. Lutwak asked if there is any
other star-shaped body that satisfy this property. We will present
a solution to a local version of this problem: if a convex body
K is closed to a unit ball and intersection body of K is equal to
K, then K is a unit ball.
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