Asymmetric norm

Definition

• Subadditivity, or the triangle inequality: p ≤ p + p for every two vectors v,w ∈ V.
• Homogeneity: p = λp for every vector v ∈ X and every non-negative real number λ ≥ 0.
• Positive definiteness: p > 0 unless v = 0.

Examples

• On the real line R, the function p given by
• In a real vector space, the Minkowski functional of a convex subset that contains the origin is defined by the formula

  Corresponce between asymmetric seminorms and convex subsets of the dual space

• positive definite if and only if contains the origin in its interior,
• degenerate if and only if is contained in a linear subspace of dimension less than, and
• symmetric if and only if.