Tropical Halfspaces

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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.