Convex body - Knowlpedia

Definition

Let $V$ be a finite-dimensional real vector space. A convex body is a nonempty compact convex set $K\subseteq V$ . The family of all convex bodies in $V$ is commonly denoted $\mathcal K(V)$ .

Operations

If $K,L\in\mathcal K(V)$ and $s,t\ge0$ , then

sK+tL=\{sx+ty:x\in K,\ y\in L\}

is again a convex body. This is the Minkowski sum operation used to polarize volume and define mixed volumes.

Topology and dual description

The Hausdorff distance metrizes convergence of convex bodies. Equivalently, $K_m\to K$ exactly when their support functions converge uniformly on compact subsets of $V^*$ , or uniformly on the unit sphere after an inner product is chosen.

Convention

Some authors require a convex body to have nonempty interior. This knowl uses the broader convention standard in valuation theory, allowing lower-dimensional compact convex sets. A statement needing full dimension must say so.

References

Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record. Relevant: §1.1.