Higher convexity for complements of tropical objects

Gromov generalized the notion of convexity for open subsets   
of R^n with hypesurface boundary, defining k-convexity, or
higher convexity and Henriques applied the same notion to 
complements of amoebas.  He conjectured that the complement 
of an amoeba of a variety of codimension k+1 is k-convex.  
I will discuss work with Mounir Nisse in which we study the 
higher convexity of complements of coamoebas and of tropical 
varieties, proving Henriques' conjecture for coamoebas and 
establishing a form of Henriques' conjecture for tropical 
varieties in some cases.