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Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

Sever Silvestru Dragomir, Jaromír J. Koliha (2000)

Applications of Mathematics

In this paper we introduce two mappings associated with the lower and upper semi-inner product ( · , · ) i and ( · , · ) s and with semi-inner products [ · , · ] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

Two-dimensional real symmetric spaces with maximal projection constant

Bruce Chalmers, Grzegorz Lewicki (2000)

Annales Polonici Mathematici

Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that λ ( V ) λ ( V n ) where V n is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that 4 / π = λ ( l ( 2 ) ) λ ( V ) for any two-dimensional real symmetric space V.

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