Convex body

Builds on Euclidean space Convex set Compact set Interior

Definition

Let $n \ge 1$ and consider Euclidean space $\R^n$ . A convex body in $\R^n$ is a subset $K \subseteq \R^n$ such that:

$K$ is convex;
$K$ is compact;
$K$ has nonempty interior.

Remarks

Example: the closed unit ball $B = \set{x \in \R^n : \abs{x} \le 1}$ . It is convex by the triangle inequality, compact because it is closed and bounded, and its interior contains the open unit ball, so it is not empty.

Non-example: the closed segment $[0, 1] \times \set{0}$ in $\R^2$ is convex and compact, but its interior in $\R^2$ is empty, so clause 3 fails.

Some authors, for example in the theory of mixed volumes, call every nonempty compact convex set a convex body and drop clause 3. With that convention, the bodies with nonempty interior are called full-dimensional.

Reading path

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Compact set concept
Convex set concept
Euclidean space object
Interior concept
Convex body definition

Tag	`YRTFS7NHM8`
URI	`m2://YRTFS7NHM8`
Kind	definition
Writing	W0 stub
MSC	52A20
Defines	Convex body
Uses	Euclidean space, Convex set, Compact set, Interior

Convex body

Definition

Remarks

Reading path

Further reading

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