Decomposition of the polyhedron from Albrecht Durer's 'Melencolia I' to a minimal pair of compact convex sets

摘要

Two pairs (A,B),(C,D) of compact convex sets are equivalent if A + D=B + C, where '+' is the Minkowski sum. In [15], the question was posed whether two equivalent minimal pairs are translates of each other. In [6], the first example of a three-dimensional minimal pair, which has an equivalent minimal pair not being the translate of the former, was given. Incidentally, the Minkowski sum of that pair resembles the famous polyhedron from Durer's 'Melancholy'. In this article, we give the decomposition of the polyhedron to a minimal pair of compact convex sets and prove that this pair belongs to a quotient class without the property of translation.