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Given a measurable set of positive measure, it is not difficult to show that if and only if is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If is small, is close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between and its convex hull in terms of .
In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...
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