Polar body

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

Let $V$ be a finite-dimensional 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 $X=\left\{x\in E^n\colon D(x)\leq 1\right\}$.