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

Question

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.

Answer 1

Hint:

$$ 0 \in f^{-1} [ (-\infty , 1) \times(-\infty , 1)\times....\times (-\infty , 1)] \subseteq Q $$

Where $f(x) = Px$ , and $P$ is a matrix whose rows are $p^{T}_i$ for $i =1,2,... N$

Clearly $f$ is continuous.