TITLE: Polarity in Discrete Geometry
QUESTION [0 upvotes]: I have no idea about how to deal with these problems. I need some advice or hint about how to approach them....... I am taking the course of Discrete Geometry maybe without solid background in linear algebra. I also want to know what should I do to compensate the gap.
(a)
Definition of polar: $A^o=\{l\in ({\bf R^d})^*|l(x)\leq1$ for all $x \in A$}
Let $A \subset B \subset {\bf R^d}$ be sets. Show that $B^o \subset A^o$.
(b) Let $A=\{x \in {\bf R^d} : a_i\cdot x \leq 1$ for $i=1,2,\cdots,m$}
Show that it's polar is $A^o=conv\{0,a_1,a_2,\cdots, a_m\}$.

REPLY [0 votes]: For part (a) suppose $a\in B^\circ$ then for every $b\in B$(also each element in $A$) $a.b\leq 1$, so $a\in A^\circ$
For part $b$ you can see answer in this book:[De_Loera,]Algebraic And geometric ideAs in the theory of discrete optimization. 
see Lemma 1.3.5.
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