A curious generalization of local uniform rotundity.

*(English)*Zbl 0578.46017The geometrical notion of rotundity in a Banach space is defined by insisting that any chord of the unit ball has its midpoint beneath the surface. The notion of uniform rotundity is described by requiring a uniform depth below the surface for midpoints of all chords of a specified length. A variety of intermediate notions have been obtained by keeping the uniform depth of midpoints requirement but only for restricted collections of chords of a specified length; for example, restricting the chords to have a common endpoint produces the notion of local uniform rotundity or restricting the chords to have a common direction produces the notion of uniform rotundity in every direction. Both of these intermediate notions are well known and have been extensively studied. In this paper, restricting both endpoints and directions of chords is seen to produce a new flavor of local uniform rotundity (specifically, the chords are restricted by requiring them to have a common endpoint as well as insisting their directions lie in a weakly compact subset not containing the zero vector). It is shown that this new notion arises in a natural way from other known generalizations of local uniform rotundity. Some basic facts about this new property are established that concern renormings, substitution spaces and quotient spaces. Finally, a similar program to generalize midpoint local uniform rotundity is considered.