1. CMB 2006 (vol 49 pp. 536)
|Measure Convex and Measure Extremal Sets |
We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.
Keywords:measure convex set, measure extremal set, face
2. CMB 1998 (vol 41 pp. 225)
|Mazur intersection properties for compact and weakly compact convex sets |
Various authors have studied when a Banach space can be renormed so that every weakly compact convex, or less restrictively every compact convex set is an intersection of balls. We first observe that each Banach space can be renormed so that every weakly compact convex set is an intersection of balls, and then we introduce and study properties that are slightly stronger than the preceding two properties respectively.
Categories:46B03, 46B20, 46A55
3. CMB 1997 (vol 40 pp. 10)
|Convex functions on Banach spaces not containing $\ell_1$ |
There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.
Categories:46A55, 46B20, 52A41