Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc $ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$.

The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0
Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential norm
Categories:47B38, 47B10, 46E40, 46E15

Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\`ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S].

Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities.

The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$.

For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$ \Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.