Mounir Nisse and
Frank
Sottile
Gromov generalized the notion of convexity for open subsets
of R 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.
Below, we show the amoeba and coamoeba of a line in (C*)3.
Amoeba of the line t -->
(t-1, t-ζ, t-ζ2), where
ζ is a primitive third root of 1.
