  As a new concept tropical halfspaces are introduced to the (linear algebraic)
geometry of the tropical semiring (R,min,+). This yields exterior descriptions
of the tropical polytopes that were recently studied by Develin and Sturmfels
in a variety of contexts. The key tool to the understanding is a newly defined
sign of the tropical determinant, which shares remarkably many properties with
the ordinary sign of the determinant of a matrix. The methods are used to
obtain an optimal tropical convex hull algorithm in two dimensions.
