Convex body

The truncated icosahedron (" soccer ball ") is a convex body in three-dimensional space

A convex body in mathematics is a geometric body that is convex and the content of which is not empty.

Definitions

A subset of -dimensional Euclidean space is called a convex body if it is convex , bounded and closed and if its interior is not empty. The convexity means that all points of the connection between two points and the body are also part of the body, that is, it applies ${\displaystyle K\subset \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle x,y\in K\Rightarrow \lambda x+(1-\lambda )y\in K}$

for everyone . The other three conditions then ensure that a convex body has only a finite extent, includes its surface and is not completely contained in a hyperplane . ${\displaystyle 0\leq \lambda \leq 1}$

A convex body is called symmetrical if for every point of the body its point mirrored at the origin is also in the body, i.e. ${\displaystyle x}$ ${\displaystyle -x}$

${\displaystyle x\in K\Rightarrow -x\in K}$

applies. A symmetrical convex body is thus centrally symmetrical with respect to the coordinate origin .

Examples

The most well-known convex bodies include the convex polyhedra , for example the regular polyhedra in three-dimensional space, of which there are five types:

• the platonic solids ,
• the Archimedes' bodies ,
• the Catalan bodies ,
• the Johnson body and
• the prisms and antiprisms .

Further examples of symmetrical convex bodies can be derived from standards , for example

• the unit sphere , ${\displaystyle K=\{x\in \mathbb {R} ^{n}\mid \|x\|_{2}\leq 1\}}$
• the unit hypercube and ${\displaystyle K=\{x\in \mathbb {R} ^{n}\mid \|x\|_{\infty }\leq 1\}}$
• the unity cross polytope , ${\displaystyle K=\{x\in \mathbb {R} ^{n}\mid \|x\|_{1}\leq 1\}}$

where is the p-norm . In general there is even a bijection between the set of symmetrical convex bodies and the set of standard spheres in the (see Minkowski functional ). ${\displaystyle \|\cdot \|_{p}}$ ${\displaystyle \mathbb {R} ^{n}}$

See also

• Minkovskian grid point theorem

literature

• Jürgen Wolfart: Introduction to number theory and algebra . Springer, 2010, ISBN 978-3-8348-9833-3 , pp. 235-236 .