Knowlpedia

Definition

Positive span

The set of all finite nonnegative combinations of a specified family of vectors.

The positive span, or conical hull, of SS in a real vector space is

pos⁡(S)={∑j=1majvj:m≥0, vj∈S, aj≥0}.\operatorname{pos}(S)=\left\{\sum_{j=1}^m a_jv_j: m\ge0,\ v_j\in S,\ a_j\ge0\right\}.

It collects all of elements of SS, including zero.

Minimality and coefficients

The positive span is the smallest convex cone containing SS: sums and nonnegative scalar multiples stay in the displayed set, and every cone containing SS contains those combinations. “Positive span” here permits zero coefficients. Requiring every coefficient of a particular finite list to be strictly positive is a separate condition. Positive span differs from convex hull because no coefficient-sum normalization is imposed.