I am curious about simplex coverings of the unit n-dimensional hypercube (or n-cube) with the following properties:
- The simplexes do not need to be regular
- The simplexes can be non-congruent (i.e. of different shapes)
- They can have vertexes outside the unit cube
- They can overlap
A trivial such covering will be a single simplex where each vertex has coordinates $v = (0, \ldots, 0, n, 0, \ldots, 0)$, thus containing the n-cube $[0,1]^n$. This is not very efficient in terms of the "amount" of the covering that is left outside the n-cube.
I am interested in the trade-off that exists between how much the covering leaves the cube, which I would quantify as $max_i ||v_i||_\infty$ where $v_i$ are the vertexes of all simplexes in the covering, and the number of simplexes required. For example, how many simplexes would be required if I wanted $max_i ||v_i||_\infty \leq n/2$.
In all of the literature I've found so far, the formulation is always much more restrictive than what I would like, for example, the simplexes are always required to be congruent or they cannot leave the n-cube. I am not sure how or where to search for this and would appreciate help in that direction.
Examples of what I found so far include:
https://link.springer.com/content/pdf/10.1007/0-387-29929-7.pdf https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1144&context=hmc_theses
