Locating the center of mass

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Mathematical definition

Let X be a locally convex topological space, and $C\subset X$ be a convex set, then the continuous linear functional $\phi :X\to \mathbb {R}$ is a supporting functional of C at the point $x_{0}$ if $\phi (x)\leq \phi (x_{0})$ for every $x\in C$.

Relation to support function

If $h_{C}:X^{*}\to \mathbb {R}$ (where $X^{*}$ is the dual space of $X$) is a support function of the set C, then if $h_{C}\left(x^{*}\right)=x^{*}\left(x_{0}\right)$, it follows that $h_{C}$ defines a supporting functional $\phi :X\to \mathbb {R}$ of C at the point $x_{0}$ such that $\phi (x)=x^{*}(x)$ for any $x\in X$.

Relation to supporting hyperplane

If $\phi$ is a supporting functional of the convex set C at the point $x_{0}\in C$ such that

$\phi \left(x_{0}\right)=\sigma =\sup _{x\in C}\phi (x)>\inf _{x\in C}\phi (x)$

then $H=\phi ^{-1}(\sigma )$ defines a supporting hyperplane to C at $x_{0}$.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.