volumenorm

Volumenorm is a term that appears in some contexts to denote a norm defined by a convex, origin-symmetric set, though in English-language literature the standard term is often Minkowski functional or gauge. When a norm is described this way, its unit ball is a balanced, convex, absorbing set around the origin, and the geometry of that set encodes the norm’s properties.

Formally, let K be a 0-symmetric convex body in a real vector space. The Minkowski functional (or gauge) p_K is defined by p_K(x) = inf{ t > 0 : x ∈ tK }. If K is balanced and absorbing, then p_K is a norm, and the unit ball of that norm is precisely K. Thus a “volumenorm” can be viewed as a norm arising from a chosen unit ball K, with the geometry of K determining the norm.

The volume of the unit ball is a geometric invariant of the norm and plays a role in comparing norms in finite-dimensional spaces. If the space has dimension n and the unit ball is scaled by a factor α, the norm scales by α and the volume of the unit ball scales by α^n. A common family of examples are the L_p norms on R^n, where the unit ball is defined by (|x1|^p + ... + |xn|^p)^{1/p} ≤ 1. The volume of this unit ball is known: Vol(B_p^n) = 2^n Γ(1+1/p)^n / Γ(1+n/p). For p = 2 this yields the familiar volume of the Euclidean unit ball.

Volumenorms are foundational in convex geometry and functional analysis, linking norm-induced geometry with volume computations and applications in optimization and analysis. See also Minkowski functional, gauge, and L_p norms.