Abstract
AbstractWe study the range of the gradients of aC1,α-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case ofC1-smooth bump functions. Finally, we give a sufficient condition on a subset ofX* so that it is the set of the gradients of aC1,1-smooth bump function. In particular, ifXis an infinite dimensional Banach space with aC1,1-smooth bump function, then any convex open bounded subset ofX* containing 0 is the set of the gradients of aC1,1-smooth bump function.
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