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Geometric containment and partial orders
 SIAM J. Discrete Math
, 1986
"... Given two solid geometric figures on the plane (eg. rectangles) we say that A fits in B (denoted A < B) if there is a translation, a rotation and (if needed) a reflection that maps A into B. Given a family F of geometric figures, we say that the relation "< " on the elements ..."
Abstract

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Given two solid geometric figures on the plane (eg. rectangles) we say that A fits in B (denoted A < B) if there is a translation, a rotation and (if needed) a reflection that maps A into B. Given a family F of geometric figures, we say that the relation &quot;< &quot; on the elements of F is reducible to vector dominance if there exists an n and a mapping f:F ÆR n such that for A,BŒF A < B iff f(A) < f(B) coordinate by coordinate. A recent result states that if F is the set of all rectangles, &quot;< &quot; is not reducible to a vector dominance relation regardless of the finite value of n. In this paper we extend this result to other families of geometric figures and to a partial order obtained from quadratic polynomials.