Editing Real coordinate space (section)

=== Convexity ===
[[File:2D-simplex.svg|thumb|The ''n''-simplex (see [[#Polytopes in Rn|below]]) is the standard convex set, that maps to every polytope, and is the intersection of the standard {{math|(''n'' + 1)}} affine hyperplane (standard affine space) and the standard {{math|(''n'' + 1)}} orthant (standard cone).]]
{{details|Convex analysis}}
In a real vector space, such as {{math|'''R'''<sup>''n''</sup>}}, one can define a convex [[cone (linear algebra)|cone]], which contains all ''non-negative'' linear combinations of its vectors. Corresponding concept in an affine space is a [[convex set]], which allows only [[convex combination]]s (non-negative linear combinations that sum to 1).

In the language of [[universal algebra]], a vector space is an algebra over the universal vector space {{math|'''R'''<sup>∞</sup>}} of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an algebra over the universal affine hyperplane in this space (of finite sequences summing to 1), a cone is an algebra over the universal [[orthant]] (of finite sequences of nonnegative numbers), and a convex set is an algebra over the universal [[simplex]] (of finite sequences of nonnegative numbers summing to 1). This geometrizes the axioms in terms of "sums with (possible) restrictions on the coordinates".

Another concept from convex analysis is a [[convex function]] from {{math|'''R'''<sup>''n''</sup>}} to real numbers, which is defined through an [[inequality (mathematics)|inequality]] between its value on a convex combination of [[point (geometry)|points]] and sum of values in those points with the same coefficients.