In 1903 Minkowski showed that, given pairwise different unit vectors
μ1, …, μm in Euclidean n-space
ℝn which span ℝn, and positive reals
μ1, …, μm such that
[sum ]mi=1μiμi = 0,
there exists a polytope P in ℝn,
unique up to translation, with outer unit facet normals μ1, …, μm and corresponding facet volumes
μ1, …, μm. This paper deals with the computational complexity of the underlying reconstruction problem,
to determine a presentation of P as the intersection of its facet halfspaces. After a natural reformulation
that reflects the fact that the binary Turing-machine model of computation is employed, it is shown that
this reconstruction problem can be solved in polynomial time when the dimension is fixed but is #ℙ-hard
when the dimension is part of the input.
The problem of ‘Minkowski reconstruction’ has various applications in image processing, and the
underlying data structure is relevant for other algorithmic questions in computational convexity.