Orthogonal Symmetries and Reflections in Banach Spaces

Abstract

Let X be a Banach space. We introduce a concept of orthogonal symmetry and reflection in X. We then establish its relation with the concept of best approximation and investigate its implication on the shape of the unit ball of the Banach space X by considering sections over subspaces. The results are then applied to the space C(I) of continuous functions on a compact set I. We obtain some nontrivial symmetries of the unit ball of C(I). We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function in X. We conclude with some suggestions for further investigations.