Terms Of Natural Properties Research Articles

Dasgupta's Detonation

Dasgupta's Detonation

Symmetrization in geometry

The concept of an i-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of i-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.

Representation of complex Banach spaces as spaces of continuous functions on a compact space

There exist many various representations of the space C(X) of continuous functions on a compact space X. Characterizations of this space in the class of real Banach spaces in terms of natural properties of normed spaces were obtained by Arens and Kelley, Jerison, and Myers. In this paper, we give a characterization of C(X) in the class of complex Banach spaces. To this end, we first describe properties of Myers functionals on a complex normed space E and define representations of the orbital function space Cσ(X) with respect to the action of a one-parameter homeomorphism group X. The real and complex cases are considered in parallel.