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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 ...

A polynomial $p \in \mathbb{R}[x_1,\dots,x_n]$ is said to be positive on a subset $S$ of $\mathbb{R}^n$ if $p(x) > 0$ for every $x \in S$. The polynomial $p$ is called nonnegative if $p(x) \ge 0$ ...