All Questions

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...

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 ...

Suppose we are given two semi-algebraic sets $S_1$ and $S_2$. Define $$ S=\{s=ps_1+(1-p)s_2: s_1\in S_1, s_2\in S_2, 0\leq p\leq 1.\} $$ $S$ is semi-algebraic. Can we bound the degree of $S$? If we ...