Support hyperplanes and polar duality of convex bodies

A hyperplane $P$ is called a support of a convex set $K$ as it has a common point with $K$ and the whole of $K$ is located on one side of $P$. Prove that it is natural to label the support planes of the unit ball in $L$ with points of the unit sphere in $L'$.

A convex set $M$ in an LTS $L$ for which the set $\left\{ x\in M\mid \forall y\in L\ \exists \varepsilon (y)\in \mathbb {R}:x+ty\in M\text{ for }\left|t\right|<\varepsilon (y)\right\}$ is nonempty is called a convex body. Convex bodies $B$ and $B'$ in $\mathbb {R}^{n}$ are said to be dual if their Minkowski functionals determine structures of dual normed spaces in $\mathbb {R}^{n}$. (We identify the vector $(a_1,\ldots ,a_n)$ with the functional $(x_1,\ldots ,x_n)\mapsto \sum_{i=1}^n a_i x_i$.) Prove that the section of $B$ by a $k$-dimensional plane $P$ is dual to the projection of $B'$ on this plane.