Polar body

2020 Mathematics Subject Classification: Primary: 52A05 [MSN][ZBL]

Let $V$ be a real vector space with inner product $\langle , \rangle$. The polar set $X^\circ$ of a subset $X$ of $V$ is $$ X^\circ = \{ y \in V : \langle x,y \rangle \le 1 \ \text{for all}\ x \in X \} \ . $$

If $K$ is a bounded convex set containing the zero element in its interior then $K^\circ$ is called the polar body of $K$ and is a compact convex neighbourhood of the origin.

The support function of $X$ may be defined in terms of the polar set by $H_X(u)=\inf\left\{\rho > 0\colon u\in \rho X^\circ \right\}$, and similarly the distance function is given by $D_X(x)=\sup\left\{\langle x,u \rangle \colon u\in X^\circ \right\}$. Given a distance function $D(x)$, the corresponding closed convex set is defined by $X=\left\{x\in E^n\colon D(x)\leq 1\right\}$.

See also: Blaschke–Santaló inequality.

References

• Arne Brøndsted, "An introduction to convex polytopes" Graduate Texts in Mathematics 90 Springer 1983 ISBN 0-387-90722-X Zbl 0509.52001
• Rolf Schneider, "Convex Bodies: The Brunn–Minkowski Theory" (2 ed.) Encyclopedia of Mathematics and its Applications 151 Cambridge University Press (2014} ISBN 1-107-60101-0 Zbl 1287.52001