2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69443As 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.18 pages, 4 figures; v3: many new references, several minor changes, new layout, renumbering of the resultsCombinatoricsMetric Geometry52B99; 06A07text