Expand all Collapse all | Results 1 - 18 of 18 |
1. CJM Online first
Une propriÃ©tÃ© de domination convexe pour les orbites sturmiennes Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$-periodic sequence of integers
($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter
(and also $N$-periodic, consequently). It is shown that, for affine
functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively
to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$-periodic
sequences), the average of $|\alpha|$ on the orbit of ${\bf x}$ is
greater than its average on the orbit of ${\bf s}$.
Keywords:suite sturmienne, domination convexe, optimisation ergodique Categories:37D35, 49N20, 90C27 |
2. CJM Online first
On the Local Convexity of Intersection Bodies of Revolution One of the fundamental results in Convex Geometry is Busemann's
theorem, which states that the intersection body of a symmetric convex
body is convex. Thus, it is only natural to ask if there is a
quantitative version of Busemann's theorem, i.e., if the intersection
body operation actually improves convexity. In this paper we
concentrate on the symmetric bodies of revolution to provide several
results on the (strict) improvement of convexity under the
intersection body operation. It is shown that the intersection body of
a symmetric convex body of revolution has the same asymptotic behavior
near the equator as the Euclidean
ball. We apply this result to show that in sufficiently high
dimension the double intersection body of a symmetric convex body of
revolution is very close to an ellipsoid in the Banach-Mazur
distance. We also prove results on the local convexity at the equator
of intersection bodies in the class of star bodies of revolution.
Keywords:convex bodies, intersection bodies of star bodies, Busemann's theorem, local convexity Categories:52A20, 52A38, 44A12 |
3. CJM Online first
Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional |
Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert-Einstein Functional The paper is centered around a new proof of the infinitesimal rigidity
of convex polyhedra. The proof is based on studying derivatives of the
discrete Hilbert-Einstein functional on the space of "warped
polyhedra" with a fixed metric on the boundary.
The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.
In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity.
We review some of the related work and discuss directions for future research.
Keywords:convex polyhedron, rigidity, Hilbert-Einstein functional, Minkowski theorem Categories:52B99, 53C24 |
4. CJM 2013 (vol 66 pp. 373)
Uniform Convexity and Bishop-Phelps-BollobÃ¡s Property A new characterization of the uniform convexity of
Banach space is obtained in the sense of Bishop-Phelps-BollobÃ¡s
theorem. It is also proved that the couple of Banach spaces $(X,Y)$
has the bishop-phelps-bollobÃ¡s property for every banach space $y$
when $X$ is uniformly convex. As a corollary, we show that the
Bishop-Phelps-BollobÃ¡s theorem holds for bilinear forms on
$\ell_p\times \ell_q$ ($1\lt p, q\lt \infty$).
Keywords:Bishop-Phelps-BollobÃ¡s property, Bishop-Phelps-BollobÃ¡s theorem, norm attaining, uniformly convex Categories:46B20, 46B22 |
5. CJM 2012 (vol 65 pp. 1236)
Higher Connectedness Properties of Support Points and Functionals of Convex Sets We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\mathrm{AR(}\sigma$-$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.
Keywords:convex set, support point, support functional, absolute retract, Leray-Schauder continuation principle Categories:46A55, 46B99, 52A07 |
6. CJM 2010 (vol 62 pp. 1293)
Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.
Keywords:toric, Fano, threefold, canonical singularities, convex polytopes Categories:14J30, 14J30, 14M25, 52B20 |
7. CJM 2010 (vol 62 pp. 827)
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces Categories:46E40, 42B25, 46B20 |
8. CJM 2009 (vol 61 pp. 566)
Convex Subordination Chains in Several Complex Variables In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.
Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination Categories:32H02, 30C45 |
9. CJM 2009 (vol 61 pp. 299)
\v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$ A set in a metric space is called a \emph{\v{C}eby\v{s}ev set} if
it has a unique ``nearest neighbour'' to each point of the space. In
this paper we generalize this notion, defining a set to be
\emph{\v{C}eby\v{s}ev relative to} another set if every point in the
second set has a unique ``nearest neighbour'' in the first. We are
interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$,
endowed with the Hausdorff metric, mainly the hyperspaces of compact
sets, compact convex sets, and strictly convex compact sets.
We present some new classes of \v{C}eby\v{s}ev and relatively
\v{C}eby\v{s}ev sets in various hyperspaces. In particular, we show
that certain nested families of sets are \v{C}eby\v{s}ev. As these
families are characterized purely in terms of containment, without
reference to the semi-linear structure of the underlying metric space,
their properties differ markedly from those of known \v{C}eby\v{s}ev
sets.
Keywords:convex body, strictly convex set, \v{C}eby\v{s}ev set, relative \v{C}eby\v{s}ev set, nested family, strongly nested family, family of translates Categories:41A52, 52A20 |
10. CJM 2007 (vol 59 pp. 614)
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators |
Preduals and Nuclear Operators Associated with Bounded, $p$-Convex, $p$-Concave and Positive $p$-Summing Operators We use Krivine's form of the Grothendieck inequality
to renorm the space of bounded linear maps acting between Banach
lattices. We
construct preduals and describe the nuclear operators
associated with these preduals for this renormed space
of bounded operators as well as for
the spaces of $p$-convex,
$p$-concave and positive $p$-summing operators acting
between Banach lattices and Banach spaces.
The nuclear operators obtained are described in
terms of factorizations through
classical Banach spaces via positive operators.
Keywords:$p$-convex operator, $p$-concave operator, $p$-summing operator, Banach space, Banach lattice, nuclear operator, sequence space Categories:46B28, 47B10, 46B42, 46B45 |
11. CJM 2007 (vol 59 pp. 85)
On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.
Keywords:Series, convergence, almost alternating, convex, continued fractions Categories:40A05, 11A55, 11B83 |
12. CJM 2007 (vol 59 pp. 3)
Holomorphic Generation of Continuous Inverse Algebras We study complex commutative Banach algebras
(and, more generally, continuous
inverse algebras) in which the holomorphic functions of a fixed $n$-tuple
of elements are dense. In particular, we characterize the compact subsets
of~$\C^n$ which appear as joint spectra of such $n$-tuples. The
characterization is compared with several established notions of holomorphic
convexity by means of approximation
conditions.
Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation Categories:46H30, 32A38, 32E30, 41A20, 46J15 |
13. CJM 2006 (vol 58 pp. 401)
On Pointwise Estimates of Positive Definite Functions With Given Support The following problem has been suggested by Paul Tur\' an. Let
$\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$
or in the torus $\TT^d$. Then, what is the largest possible value
of the integral of positive definite functions that are supported
in $\Omega$ and normalized with the value $1$ at the origin? From
this, Arestov, Berdysheva and Berens arrived at the analogous
pointwise extremal problem for intervals in $\RR$. That is, under
the same conditions and normalizations, the supremum of possible
function values at $z$ is to be found for any given point
$z\in\Omega$. However, it turns out that the problem for the real
line has already been solved by Boas and Kac, who gave several
proofs and also mentioned possible extensions to $\RR^d$ and to
non-convex domains as well.
Here we present another approach to the problem, giving the
solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we
elaborate on the fact that the problem is essentially
one-dimensional and investigate non-convex open domains as well.
We show that the extremal problems are equivalent to some more
familiar ones concerning trigonometric polynomials, and thus find
the extremal values for a few cases. An analysis of the
relationship between the problem for $\RR^d$ and that for $\TT^d$
is given, showing that the former case is just the limiting case
of the latter. Thus the hierarchy of difficulty is established, so
that extremal problems for trigonometric polynomials gain renewed
recognition.
Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems Categories:42B10, 26D15, 42A82, 42A05 |
14. CJM 2003 (vol 55 pp. 1000)
Some Convexity Results for the Cartan Decomposition In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$
where $a(g)$ is the abelian part in the Cartan decomposition of
$g$. This is exactly the support of the measure intervening in the
product formula for the spherical functions on symmetric spaces of
noncompact type. We give a simple description of that support in
the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$,
$\mathbf{C}$ or $\mathbf{H}$. In particular, we show that
$\mathcal{S}$ is convex.
We also give an application of our result to the description of
singular values of a product of two arbitrary matrices with
prescribed singular values.
Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values Categories:43A90, 53C35, 15A18 |
15. CJM 2002 (vol 54 pp. 916)
ConvexitÃ©, complÃ¨te monotonie et inÃ©galitÃ©s sur les fonctions zÃªta et gamma sur les fonctions des opÃ©rateurs de Baskakov et sur des fonctions arithmÃ©tiques |
ConvexitÃ©, complÃ¨te monotonie et inÃ©galitÃ©s sur les fonctions zÃªta et gamma sur les fonctions des opÃ©rateurs de Baskakov et sur des fonctions arithmÃ©tiques We give optimal upper and lower bounds for the function
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ for $x\geq 0$ and $s>1$. These
bounds improve the standard inequalities with integrals. We deduce from them
inequalities about Riemann's $\zeta$ function, and we give a conjecture
about the monotonicity of the function
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. Some applications concern the
convexity of functions related to Euler's $\Gamma$ function and optimal
majorization of elementary functions of Baskakov's operators. Then, the
result proved for the function $x\mapsto x^{-s}$ is extended to completely
monotonic functions. This leads to easy evaluation of the order of the
generating series of some arithmetical functions when $z$ tends to 1. The
last part is concerned with the class of non negative decreasing convex
functions on $]0,+\infty[$, integrable at infinity.
Nous prouvons un encadrement optimal pour la quantit\'e
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ pour $x\geq 0$ et $s>1$, qui
am\'eliore l'encadrement standard par des int\'egrales. Cet encadrement
entra{\^\i}ne des in\'egalit\'es sur la fonction $\zeta$ de Riemann, et
am\`ene \`a conjecturer la monotonie de la fonction
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. On donne des applications \`a
l'\'etude de la convexit\'e de fonctions li\'ees \`a la fonction $\Gamma$
d'Euler et \`a la majoration optimale des fonctions \'el\'ementaires
intervenant dans les op\'erateurs de Baskakov. Puis, nous \'etendons aux
fonctions compl\`etement monotones sur $]0,+\infty[$ les r\'esultats \'etablis
pour la fonction $x\mapsto x^{-s}$, et nous en d\'eduisons des preuves
\'el\'ementaires du comportement, quand $z$ tend vers $1$, des s\'eries
g\'en\'eratrices de certaines fonctions arithm\'etiques. Enfin, nous
prouvons qu'une partie du r\'esultat se g\'en\'eralise \`a une classe de
fonctions convexes positives d\'ecroissantes.
Keywords:arithmetical functions, Baskakov's operators, completely monotonic functions, convex functions, inequalities, gamma function, zeta function Categories:26A51, 26D15 |
16. CJM 2001 (vol 53 pp. 470)
Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convex-analytic tools
for such symmetric functions, of interest in interior-point methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 |
17. CJM 2000 (vol 52 pp. 141)
Numerical Ranges Arising from Simple Lie Algebras A unified formulation is given to various generalizations of the
classical numerical range including the $c$-numerical range,
congruence numerical range, $q$-numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.
Keywords:numerical range, convexity, inclusion relation Categories:15A60, 17B20 |
18. CJM 1999 (vol 51 pp. 250)
Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'e-Kuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized second-order derivability of
convexly composite functions.
Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 |