Star-shaped and convex sets in the Heisenberg group

Doaa Filali (Cardiff, UK)

Star-shaped sets play a signifcant role in analysis and PDEs. In particular convex sets are star-shaped with respect to the all interior points. In the Euclidean case there are several equivalent definitions for star-shaped sets. This is not true in more degenerate geometries. For example in the Heisenberg group we generalised several Euclidean definitions for starshapedness and they turned to be not equivalent. For example we looked at the definition of strong star-shaped sets (related to the dilations) and the definition of weak star-shaped sets (related to the horizontal line segments). We construct counterexamples showing that the two definitions are not equivalent. While the first definition is important for some PDE application, the second weaker definition is key for convexity. In fact weak star-shaped with respect to all interior points is equivalent to the horizontal convexity (and all other equivalent notions known in the Heisenberg group). Beside that we study the relation between many different notions of star-shaped sets and their relations with convexity and convex functions. These results have been applied in the study of geometrical properties for level sets of nonlinear subelliptic PDEs.

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