On polynomial images of a closed ball

In this work we approach the problem of determining which (compact) semialgebraic subsets of ${\mathbb R}^n$ are images under polynomial maps $f:{\mathbb R}^m\to{\mathbb R}^n$ of the closed unit ball $\overline{{\mathcal B}}_m$ centered at the origin of some Euclidean space ${\mathbb R}^m$ and that of estimating (when possible) which is the smallest $m$ with this property. Contrary to what happens with the images of ${\mathbb R}^m$ under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension $n$) are polynomial images of the closed unit ball in ${\mathbb R}^n$. The previous examples (and other basic ones proposed in the article) provide a large family of `$n$-bricks' and we find necessary and sufficient conditions to guarantee that a finite union of `$n$-bricks' is again a polynomial image of the closed unit ball either of dimension $n$ or $n+1$. In this direction, we prove: {\em A finite union ${\mathcal S}$ of $n$-dimensional convex polyhedra is the image of the $n$-dimensional closed unit ball $\overline{{\mathcal B}}_n$ if and only if ${\mathcal S}$ is connected by analytic paths}. The previous result can be generalized using the `$n$-bricks' mentioned before and we show: {\em If ${\mathcal S}_1,\ldots,{\mathcal S}_\ell\subset{\mathbb R}^n$ are `$n$-bricks', the union ${\mathcal S}:=\bigcup_{i=1}^\ell{\mathcal S}_i$ is the image of the closed unit ball $\overline{{\mathcal B}}_{n+1}$ of ${\mathbb R}^{n+1}$ under a polynomial map $f:{\mathbb R}^{n+1}\to{\mathbb R}^n$ if and only if ${\mathcal S}$ is connected by analytic paths}.