I am working on determining the spatial information of a set of $M$ points in $N$-dimensional space. It is well-known that the construction of Delaunay triangulation is expensive in high dimensional space. Thus, I would like to only find the largest simplex of $N$-dimensional Delaunay triangulation (DT). I am going to sample a new point inside that simplex by each iteration, which means that the largest simplex would become smaller and smaller by iteration.
Is there any idea or references to computing the largest simplex in DT? It may not be exactly the largest simplex after computing DT. But it should at least (1) approximate the largest simplex within a tolerance; (2) the largest simplex that this method computed should become smaller and smaller by each iteration.