Orthant

In geometry, an orthant or hyperoctant is the analogue in "n"-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in "n"-dimensions can be considered the intersection of "n" mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2"n" orthants in "n"-dimensional space.
More specifically, a closed orthant in R"n" is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities:
where each ε"i" is +1 or −1.
Similarly, an open orthant in R"n" is a subset defined by a system of strict inequalities
where each ε"i" is +1 or −1.
By dimension:
John Conway defined the term "n"-orthoplex from orthant complex as a regular polytope in "n"-dimensions with 2"n" simplex facets, one per orthant.
The nonnegative orthant is the generalization of the first quadrant to "n"-dimensions and is important in many constrained optimization problems.