# Prove that the polar set of a convex set contains the origin

Let the polytope $$P = \text{conv}\{p_1,\ldots,p_N\} \subset \mathbb{R}^n$$ and the polar set defined by $$Q = \{x\in \mathbb{R}^n : p_i^Tx \le 1, \forall i \in \{1,\ldots,N\}\}.$$ Prove that the origin is in the interior of Q i.e., $$0\in \text{int}(Q)$$.

It seems to be straightforward but I don't see it clearly. Thanks.

$$0 \in f^{-1} [ (-\infty , 1) \times(-\infty , 1)\times....\times (-\infty , 1)] \subseteq Q$$
Where $$f(x) = Px$$ , and $$P$$ is a matrix whose rows are $$p^{T}_i$$ for $$i =1,2,... N$$
Clearly $$f$$ is continuous.