Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as

$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$

with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a finite index set $I$.

Question: Is the polar dual $C^\circ$ of a semi-algebraic convex body again semi-algebraic?

Where polar dual means

$$C^\circ := \{y\in\Bbb R^n\mid \langle x,y\rangle\le 1\text{ for all $x\in\Bbb R^n$}\}.$$