This is an announcement for the paper "On the divergence of series of
the form sum_{k=1}^infty||A_k x||^p" by Ivan Feshchenko.
Abstract: Let {A} be a system of operators. With any element x we
associate the set of elements {Ax}. We study conditions under which
there exists an element x such that the sum of p-th powers of norms of
the elements {Ax} is equal to infinity.
Archive classification: math.FA
Mathematics Subject Classification: 40H05, 46B20, 47A05
Remarks: 9 pages, submitted to Studia Mathematica
Submitted from: ivanmath007(a)gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.1863
or
http://arXiv.org/abs/1208.1863
This is an announcement for the paper "Some affine invariants revisited"
by Alina Stancu.
Abstract: We present several sharp inequalities for the $SL(n)$ invariant
$\Omega_{2,n}(K)$ introduced in our earlier work on centro-affine
invariants for smooth convex bodies containing the origin. A connection
arose with the Paouris-Werner invariant $\Omega_K$ defined for convex
bodies $K$ whose centroid is at the origin. We offer two alternative
definitions for $\Omega_K$ when $K \in C^2_+$. The technique employed
prompts us to conjecture that any $SL(n)$ invariant of convex bodies with
continuous and positive centro-affine curvature function can be obtained
as a limit of normalized $p$-affine surface areas of the convex body.
Archive classification: math.FA
Mathematics Subject Classification: 52A40, 52A38
Remarks: 15 pages
Submitted from: stancu(a)mathstat.concordia.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.0783
or
http://arXiv.org/abs/1208.0783
This is an announcement for the paper "Locally decodable codes and the
failure of cotype for projective tensor products" by Jop Briet, Assaf
Naor, and Oded Regev.
Abstract: It is shown that for every $p\in (1,\infty)$
there exists a Banach space $X$ of finite cotype such that
the projective tensor product $\ell_p\tp X$ fails to have
finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$
satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then
$\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This
is a proved via a connection to the theory of locally decodable codes.
Archive classification: math.FA cs.CC
Submitted from: odedr(a)cs.tau.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.0539
or
http://arXiv.org/abs/1208.0539
This is an announcement for the paper "Rotation equivariant Minkowski
valuations" by Rolf Schneider and Franz E. Schuster.
Abstract: The projection body operator \Pi, which associates with every
convex body in Euclidean space Rn its projection body, is a continuous
valuation, it is invariant under translations and equivariant under
rotations. It is also well known that \Pi\ maps the set of polytopes in
Rn into itself. We show that \Pi\ is the only non-trivial operator with
these properties.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52A20, 52B11, 52B45
Citation: Int. Math. Res. Not. 2006, Art. ID 72894, 20 pp
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7279
or
http://arXiv.org/abs/1207.7279
This is an announcement for the paper "The Steiner formula for Minkowski
faluations" by Lukas Parapatits and Franz E. Schuster.
Abstract: A Steiner type formula for continuous translation invariant
Minkowski valuations is established. In combination with a recent result
on the symmetry of rigid motion invariant homogeneous bivaluations, this
new Steiner type formula is used to obtain a family of Brunn-Minkowski
type inequalities for rigid motion intertwining Minkowski valuations.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52B45, 52A40
Citation: Adv. in Math. 230 (2012), 978-994
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7276
or
http://arXiv.org/abs/1207.7276
This is an announcement for the paper "The Sine transform of isotropic
measures" by Gabriel Maresch and Franz E. Schuster.
Abstract: Sharp isoperimetric inequalities for the sine transform of even
isotropic measures are established. The corresponding reverse inequalities
are obtained in an asymptotically optimal form. These new inequalities
have direct applications to strong volume estimates for convex bodies
from data about their sections or projections.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52A20, 52A41, 53Cxx
Citation: Int. Math. Res. Not. IMRN 2012, 717–739
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7266
or
http://arXiv.org/abs/1207.7266
This is an announcement for the paper "GL(n) contravariant Minkowski
valuations" by Franz E. Schuster and Thomas Wannerer.
Abstract: A complete classification of all continuous GL(n) contravariant
Minkowski valuations is established. As an application we present a
family of sharp isoperimetric inequalities for such valuations which
generalize the classical Petty projection inequality.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52B45, 52A20, 52A40
Citation: Trans. Amer. Math. Soc. 364 (2012), 815–826
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7256
or
http://arXiv.org/abs/1207.7256
This is an announcement for the paper "Crofton measures and Minkowski
valuations" by Franz E. Schuster.
Abstract: A description of continuous rigid motion compatible Minkowski
valuations is established. As an application, we present a Brunn-Minkowski
type inequality for intrinsic volumes of these valuations.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52B45, 43A90, 52A40
Citation: Duke Math. J. 154 (2010), 1–30
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7254
or
http://arXiv.org/abs/1207.7254
This is an announcement for the paper "Convolutions and multiplier
transformations of convex bodies" by Franz E. Schuster.
Abstract: Rotation intertwining maps from the set of convex bodies
in Rn into itself that are continuous linear operators with respect
to Minkowski and Blaschke addition are investigated. The main focus
is on Blaschke-Minkowski homomorphisms. We show that such maps are
represented by a spherical convolution operator. An application of this
representation is a complete classification of all even Blaschke-Minkowski
homomorphisms which shows that these maps behave in many respects similar
to the well known projection body operator. Among further applications
is the following result: If an even Blaschke-Minkowski homomorphism
maps a convex body to a polytope, then it is a constant multiple of the
projection body operator.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52A20, 43A90, 52A40
Citation: Trans. Amer. Math. Soc. 359 (2007), 5567–5591
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7252
or
http://arXiv.org/abs/1207.7252
This is an announcement for the paper "An arithmetic proof of John’s
ellipsoid theorem" by Peter M. Gruber and Franz E. Schuster.
Abstract: Using an idea of Voronoi in the geometric theory of positive
definite quadratic forms, we give a transparent proof of John’s
characterization of the unique ellipsoid of maximum volume contained
in a convex body. The same idea applies to the ‘hard part’ of a
generalization of John’s theorem and shows the difficulties of the
corresponding ‘easy part’.
Archive classification: math.MG math.DG math.FA
Mathematics Subject Classification: 52A21, 46B07, 52A27
Citation: Arch. Math. (Basel) 85 (2005), 82–88
Submitted from: franz.schuster(a)tuwien.ac.at
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.7246
or
http://arXiv.org/abs/1207.7246