A closed coordinate-sign region of Euclidean space. The orthants cover the space. An orthant contains no nontrivial affine line: for a nonzero direction, some coordinate eventually violates its sign restriction in one direction. Adding orthant inequalities to a nonempty linear polyhedron therefore permits construction of a basic feasible solution even when the original polyhedron has no vertex.
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In mathematics, particularly in the fields of geometry and topology, an "orthant" refers to a generalization of quadrants in higher-dimensional spaces. Specifically, it denotes a portion of a Cartesian coordinate system, defined by the signs of the coordinates. For example, in a two-dimensional space (2D), the space is divided into four quadrants based on the signs of the x and y coordinates.