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

$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 1\}$$

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

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

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Secondary questions: If the answer to my question is No, what can be said about the class of semi-algebraic convex bodies for which this is true? If Yes, how would one compute a representation as in $(*)$ for $C^\circ$? I can imagine that there exists quite some literature in this direction (real algebraic geoemtry? algebraic convex geometry? semi-algebraic geometry?), but I could not find anything specific and I am happy about any keyword or reference.