Upper estimates for Gâteaux differentiability of bump functions in Orlicz–Lorentz spaces.
Upper estimates on self-perimeters of unit circles for gauges
Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.
Using boundaries to find smooth norms
The aim of this paper is to present a tool used to show that certain Banach spaces can be endowed with smooth equivalent norms. The hypothesis uses particular countable decompositions of certain subsets of , namely boundaries. Of interest is that the main result unifies two quite well known results. In the final section, some new corollaries are given.