Speaker
Description
We focus on decomposability and extremality properties of nonlocal perimeters. Two archetypal types of these are the Gagliardo perimeter based on the eponymous seminorms and the nonlocal distributional Caccioppoli perimeter, both which can be considered with with finite and infinite interaction ranges.
A nonlocal notion of indecomposability associated to these perimeters is introduced, and we prove that it can be characterized solely in terms of the interaction range or horizon $\varepsilon$. Utilizing this, we show that it is possible to uniquely decompose a set into its $\varepsilon$-connected components, establishing a nonlocal analogue of the decomposition theorem of Ambrosio, Caselles, Masnou and Morel. This result is shown to apply also to Minkowski-type and adversarial perimeters.
Moreover, the extreme points of the balls induced by the Gagliardo and nonlocal total variation seminorm are identified, which naturally correspond to the two nonlocal perimeters. Surprisingly, while the extreme points in the former case are normalized indicator functions of $\varepsilon$-simple sets, akin to the classical TV-ball, in the latter case they are instead obtained from a nonlocal transformation applied to the extreme points of the TV-ball.