Averaging distances in finite dimensional normed spaces and John's ellipsoid

A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all in the unit sphere of there is some in the unit sphere of such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , then . This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of .

     

Angles in normed spaces

The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.     

New sequence spaces in multiple normed spaces

In this work, using lacunary sequences and the notion of ideal convergence we define and examine new sequence spaces with respect to a sequence of modulus functions in n-normed linear spaces. Further, the definition of I_θ-convergence in n-normed linear spaces and some related results are given.     

Star-shaped sets in normed spaces

We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces.     

Compactness in asymmetric normed spaces

A systematic study of precompact and compact subsets on asymmetric normed linear spaces is developed, centering our attention in the case of linear lattices with an asymmetric norm.     

Sheltered points in normed spaces

Summary If X is a Hilbert space and V is a subspace, given x ε X/V the best approximation II_v(x) to x from V can be constructed in the following way: we consider a ball B_r centered at x and intersecting V, then we take the (Chebyshev) center of B_r ∩ V. In Banach spaces the centers in V of B_r ∩ V (which depend on r) are related to a different map of ? approximation ?. Here we introduce the notions of ? sheltered point ? and ? shelter ? of a bounded set, and we obtain in Banach spaces information about II_v(x) starting from the shelter of B_r ∩ V. The notion of sheltered point turns out to be of some interest by itself, since translates problems of simultaneous worst approximation. Entrata in Redazione il 1° giugno 1977. Work performed under the auspices of the CNR (Consiglio Nazionale delle Ricerche).     

Probabilistic normed Riesz spaces

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.     

Pyramids in three-dimensional normed spaces

It is proved that vertex sets of Euclidean tetrahedra of special type can be isometrically embedded in an arbitrary three-dimensional normed space. In particular, this holds true for every regular triangular pyramid. Bibliography: 9 titles.     

A characterization of normed spaces

We prove that the metric characterization of real normed spaces obtained by T. Oikhberg and H. Rosenthal can be obtained without a continuity assumption provided that the space is at least two-dimensional. In order to get this improvement we first need to understand the exceptional one-dimensional case.     

Direct sum of normed spaces

Khar'kov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 186–189, January–February, 1991.     

Minimality of planes in normed spaces

We prove that a region in a two-dimensional affine subspace of a normed space V has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to Λ² V. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon.     

Statistical continuity in probabilistic normed spaces

In this study, we investigate the statistical continuity in a probabilistic normed space. In this context, the statistical continuity properties of the probabilistic norm, the vector addition and the scalar multiplication are examined.     

Best coapproximation in normed linear spaces

We consider, in normed linear spaces, a kind of approximation by elements of linear subspaces, introduced byC. Franchetti andM. Furi [5], which we call best coapproximation. We obtain some results on characterization and existence of elements of best coapproximation in arbitrary normed linear spaces and in spaces of continuous functions. We give some characterizations of strict convexity in terms of best coapproximation and we study some properties of the setvalued operators of best coapproximation.Work performed partially under the auspices of the GNAFA (National Group for Functional Analysis and its Applications) of the CNR (National Research Council of Italy)     

Nonlinear programming in normed linear spaces

Necessary conditions in the form of multiplier rules are given for a function to have a constrained minimum. First-order differentiability conditions are imposed, and various combinations of set, equality, and inequality constraints are considered in arbitrary normed linear spaces.This paper is based upon part of the author's doctoral dissertation at Ohio University, Athens, Ohio.