Support function of a convex body

Definition

For a convex body $K\subseteq V$ , its support function is

h_K:V^*\longrightarrow\mathbb R, \qquad h_K(\ell)=\sup_{x\in K}\ell(x).

After choosing an inner product and identifying $V^*\cong V$ , this becomes $h_K(y)=\sup_{x\in K}\langle x,y\rangle$ .

Characterization

The support function is finite, continuous, positively homogeneous, and subadditive. Conversely, every finite continuous sublinear function on $V^*$ is the support function of a unique convex body. Thus $h_K$ determines $K$ .

Operations

For $s,t\ge0$ ,

h_{sK+tL}=s h_K+t h_L.

Translations add a linear function: $h_{K+a}(\ell)=h_K(\ell)+\ell(a)$ . If $K\cup L$ is convex, then

h_{K\cup L}=\max(h_K,h_L), \qquad h_{K\cap L}=\min(h_K,h_L).

The max/min identities connect Hessian measures to valuations.

Plurisubharmonic viewpoint

Because $h_K$ is convex, it is subharmonic and, on complex, quaternionic, or the octonionic plane, belongs to the corresponding plurisubharmonic class. Its Hessian measure can therefore be used in the pluripotential construction of valuations.

References

Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record. Relevant: §1.7.