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Search: MSC category 46B20 ( Geometry and structure of normed linear spaces )

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26. CMB 2002 (vol 45 pp. 232)

Ji, Min; Shen, Zhongmin
On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry. A strongly convex indicatrix in a vector space is a strongly convex hypersurface. It admits a Riemannian metric and has a distinguished invariant---(Cartan) torsion. We prove the existence of non-trivial strongly convex indicatrices with vanishing mean torsion and discuss the relationship between the mean torsion and the Riemannian curvature tensor for indicatrices of Randers type.

Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35

27. CMB 2002 (vol 45 pp. 3)

Azagra, D.; Dobrowolski, T.
Real-Analytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinite-dimensional Banach space $X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an infinite-dimensional Banach space and $F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and $X/F$ is infinite-dimensional, then $X$ and $X \setminus F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the $n$-torus on certain Banach spaces.

Categories:46B20, 58B99

28. CMB 1999 (vol 42 pp. 221)

Liu, Peide; Saksman, Eero; Tylli, Hans-Olav
Boundedness of the $q$-Mean-Square Operator on Vector-Valued Analytic Martingales
We study boundedness properties of the $q$-mean-square operator $S^{(q)}$ on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2 \leq q < \infty$. We establish that a.s. finiteness of $S^{(q)}$ for every bounded $E$-valued analytic martingale implies strong $(p,p)$-type estimates for $S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function $S^{(q)}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly PL-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the $L^p$-boundedness of the usual square-function on scalar-valued analytic martingales.

Categories:46B20, 60G46

29. CMB 1999 (vol 42 pp. 118)

Rao, T. S. S. R. K.
Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$
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.

Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-ideals
Categories:46B20, 46E40

30. CMB 1998 (vol 41 pp. 279)

Acosta, María D.; Galán, Manuel Ruiz
New characterizations of the reflexivity in terms of the set of norm attaining functionals
As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space $X$ is very smooth or its bidual satisfies the $w^{\ast }$-Mazur intersection property, then either $X$ is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if $X$ has the Mazur intersection property and so, if the norm of $X$ is Fr\'{e}chet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

Categories:46B04, 46B10, 46B20

31. CMB 1998 (vol 41 pp. 145)

Fry, R.
Smooth partitions of unity on Banach spaces
It is shown that if a Banach space $X$ admits a $C^k$-smooth bump function, and $X^{*}$ is Asplund, then $X$ admits $C^k$-smooth partitions of unity.

Category:46B20

32. CMB 1998 (vol 41 pp. 225)

Vanderwerff, Jon
Mazur intersection properties for compact and weakly compact convex sets
Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.

Categories:46B03, 46B20, 46A55

33. CMB 1997 (vol 40 pp. 10)

Borwein, Jon; Vanderwerff, Jon
Convex functions on Banach spaces not containing $\ell_1$
There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.

Categories:46A55, 46B20, 52A41
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