I'm trying to understand Roudneff's proof of a theorem of Tverberg, in discrete and convex geometry. The proof is given here: https://arxiv.org/pdf/1712.06119.pdf (proof on pg3).
Questions:
(1) Why does there exist $y_j$ so that $\mathrm{dist}(z, \mathrm{conv} ~X_j) = \|x - y_j\|$? Is it because $\mathrm{conv}~X_j$ is a compact set (it is the image of a continuous function applied to a compact set) and hence $\|x - y\|$ attains a minimum over the convex hull of $X_j$ by EVT?
(2) Why is the second to last sentence of the third to final paragraph on page 3 true? Why does that function attain minimum at the same point?
(3) Why is the second to last sentence of second to final paragraph of page 3 true? Why is that inner product > 0? (why is this implied by what is in the parentheses)?
(4) Why are the first and third sentences of the last paragraph of page 3 true? (In particular, why is the first sentence a result of the general position assumption and why does inner product > 0 imply we can reduce mu?)
(5) Why can we make the assumption that $X$ lies in general position?