On the linear polarization constants of finite dimensional spaces

We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces : they behave as if and as if . For we get the asymptotic behaviour up to a logarithmic factor.     

On finite dimensional subspaces of Banach spaces

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice.     

On extending real functions to finite dimensional algebras

In a finite dimensional real algebra, we define functional calculus for real functions. The analytic hypothesis is replaced by a differentiable hypothesis. High order differentibility is essentially necessary and sufficient to our discussions. Moreover, we give some examples. Partially supported by NSF of China The authors gratefully acknowledge the support of K. C. Wong Education Foundation, Hong Kong     

Preservers of generalized orthogonality on finite dimensional real vector and projective spaces

We describe the general form of bijective orthogonality preserving maps on ? ⁿ equipped with a pair of generalized indefinite inner products. The relations between the projective space and vector space versions of this result are examined and an example is given showing that the hypotheses of our main theorem are essential.     

On a class of finite linear spaces with few lines

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421–423) states that a finite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing finite linear spaces for which the difference between the number b of lines and the number v of points is assigned.

In this paper finite linear spaces with b−vm, m being the minimum number of lines on a point, are characterized.     

On totally Fenchel unstable functions in finite dimensional spaces

We give an answer to the Problem 11.6 posed by Stephen Simons in his book “From Hahn-Banach to Monotonicity”: Do there exist a nonzero finite dimensional Banach space and a pair of extended real-valued, proper and convex functions which is totally Fenchel unstable? The answer is negative.     

On existence of cone-maximal points in real topological linear spaces

A sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points. The condition becomes a necessary one in a topological linear space with a countable local base, that is, if the space is pseudometrizable. The paper extends known results to infinite dimensions and we answer Corley’s question in the affirmative with the exception of a pathological case.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

Vector packing in finite dimensional vector spaces

Let m be an integer ? 2. The effect of crowding m unit vectors x₁,…,x_m into the real Euclidean space R_n of n dimensions is investigated. In particular, several upper bounds for the quantity min_i≠j∥x_i ? x_j∥ are obtained. These are simpler than any previously known and, at least in some cases, almost as sharp. The results have application to the so-called maximum-dispersal (or “misanthrope”) problem, an open problem recently popularized by Klee.     

Differential quasi-variational inequalities in finite dimensional spaces

In this paper, we introduce and study a new class of differential quasi-variational inequalities in finite dimensional Euclidean spaces. First, we prove existence theorems for Carathéodory weak solutions of the differential quasi-variational inequalities under various conditions. Furthermore, we establish a convergence result on Euler time-dependent procedure for solving the initial-value differential set-valued variational inequalities.     

Constructing frames for finite dimensional Hilbert spaces

In this paper, we use the characterization of frames, introduced in [W. Rolli, A characterization of frames for finite dimensional Hilbert spaces, preprint], to construct frames for finite dimensional Hilbert spaces. Furthermore this approach allows for the construction of frames with prescribed properties.     

On a class of real Banach spaces

The structure theory for simplex spaces is extended to arbitrary real Banach spaces with L¹-duals. This research was supported in part by the National Science Foundation.     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.