It sounds such as you’ve barely misunderstood Welzl’s minimal enclosing sphere algorithm.
It does not declare {that a} level $P_k$ outdoors an intermediate Minimal Enclosing Sphere of factors $P_1-P_{k-1}$ have to be on the boundary of the closing sphere.
It says that such some extent have to be on the boundary of the minimal enclosing sphere of factors $P_1 – P_k$, which is a considerably much less demanding declare. It might or will not be on the boundary of the minimal enclosing sphere of factors $P_1 – P_n$ for $n > ok$ — the algorithm makes no assumption about this.
This implies if we selected any sphere to surround $P_1 – P_k$ for which $P_k$ is in its inside, then we expanded our sphere “too far” – we overshot $P_k$ and included some empty area on the far aspect of it. If we backtrack a bit, shrinking our sphere to exclude that vacant area whereas maintaining all factors $P_1 – P_k$ lined, we’ll finally hit some extent the place the boundary hits $P_k$.
The smallest sphere enclosing all factors $P_1-P_k$ with $P_k$ on its boundary is essentially smaller than any different sphere containing the identical factors with $P_k$ in its inside (on condition that $P_k$ will not be in $textual content{MES}(P_1-P_{k-1})$)
I’ve a C# implementation of the algorithm written up on this reply. You’ll be able to see how the boundary factors chosen by the algorithm apply to deeper recursive calls, however do not bubble as much as the mum or dad calls from earlier within the tree.
One other approach to consider that is to replicate on the truth that it is a dynamic programming algorithm, that means it exploits optimum substructure: The optimum answer to some drawback of dimension $n$ is constructed out of options to sub-problems of dimension $ok < n$. Which means in the event you agree an assumption is legitimate for the ultimate step of the algorithm, it have to be legitimate for all steps earlier than that too. Why? As a result of the second-last step of the algorithm is itself the “closing step” of the algorithm when working on some smaller knowledge set, and we already conceded that the belief is legitimate for the ultimate step. QED. 😜
