The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: Combinatorial analysis

Letf1, ...,fmbe (partially defined) piecewise linear functions ofd variables whose graphs consist ofn d-simplices altogether. We show that the maximal number ofd-faces comprising the upper envelope (i.e., the pointwise maximum) of these functions isO(ndα(n)), whereα(n) denotes the inverse of the Ackermann function, and that this bound is tight in the worst case. If, instead of the upper envelope, we consider any single connected componentC enclosed byn d-simplices (or, more generally, (d − 1)-dimensional compact convex sets) in ℝd+1, then we show that the overall combinatorial complexity of the boundary ofC is at mostO(nd+1−ɛ(d+1)) for some fixed constantɛ(d+1)>0.

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