Piecewise-rational retractions onto closed, convex, semi-algebraic sets with interior-synopsis (Q1061799)

Let K be an ordered field contained in a real closed order-extension field R. A semi-algebraic set of K is defined as a boolean combination of sets of the form \(\{x\in K^ n| f(x)>0\}\) with \(f\in K[X_ 1,...,X_ n]\). A piecewise rational function on A is a function f such that there exists a partition of A in semi-algebraic sets \(A_ i\) such that the restriction of f to each \(A_ i\) is rational. A sketch of the proof of the following result is given: Let K be archimedean. Let \(W\subset {\mathbb{R}}^ n\) be a closed convex set with non-empty interior. Then W is a piecewise rational neighbourhood retract. One can hope to use this result to get a continuous solution to Hilbert 17-th problem for more general fields. The proof for real closed fields by the author used semi-algebraic retractions.