Let $C\subset\mathbb{R}^N$ be a compact convex subset. Our goal is to recover the best approximation possible to $C$ from its projections onto rays. To put it in specific terms:
To us a ray is a 1-dimensional linear subspace $V$ of $\mathbb{R}^N$ (or just an element of $\mathbb{P}^{N-1}$ if you wish). We denote by $P_V:\mathbb{R}^N\rightarrow V$ the projection onto $V$.
Now, given a finite set of (pairwise different) rays $V_1,\cdots,V_n$ take $S:=\bigcap_{i=1}^nP_V^{-1}(P_V(C))$ as an approximation of $C$ (we have $C\subset S$ and, if we let $n$ increase, $S$ becomes closer to $C$ in the sense that its volume is reduced).
Suppose that we don't know much about $C$, but that it is easy for us to check whether a point is in $S$ or not. What is the wisest way to choose the rays $V_1,\ldots,V_n$ so that we minimise the volume of $S$?
We are aware that using some facts about $C$ it is possible to choose the rays wisely, but as we said, don't assume we know much about it. We are looking for the best generic way of choosing the rays. Taking that into account our first approach is to think of a way of choosing the rays as evenly distributed as possible. One possible way to do it would be the following:
- Step 1: Choose $[1:0:\cdots:0],\ldots,[0:\cdots:0:1]$ as an initial set of $N$ rays. Let's call this set $R_1$.
- Step 2: Take $R_2$ the set of $2^{N-1}$ "barycentric" rays formed by the elements of $R_1$ (e.g., for $N=3$, $R_2=\{[1:1:1],[-1:1:1],[1:-1:1],[1:1:-1]\}$).
- Step $K$: Take $R_K$ as the barycentric rays formed by the elements of $R_1\cup\cdots\cup R_{K-1}$.
- Final Step: after a finite number $K$ of steps (determined by your computational capacity, say) take $R_1\cup\cdots\cup R_K$ as the set of rays to compute $S$.
Given the described problem, is this the best way to choose the rays or is there a better way to do so?