Determining a polytope by Radon partitions
Marilyn Breen · Pacific Journal of Mathematics · 1972
In an extension of the classical Radon theorem, Hare and Kenelly have introduced the concept of a primitive partition, allowing* a reduction to minimal subsets which still possess the necessary intersection property.Here it is proved that primitive partitions in the vertex set P of a polytope reveal the subsets of P which give rise to faces of conv P, thus determining the combinatorial type of the polytope.Furthermore, the polytope may be reconstructed from various subcollections of the primitive partitions.2* Preliminary results* Throughout, | P | denotes the cardinality of P. If P is a set of points in R d , A U B is a Radon partition for P iff P = A\J B,AΓ) B = 0, and conv A Π conv B Φ 0. Each of A and B is called half a partition for P and each element of A is said to oppose B in the partition.The Radon theorem says that for P^R d having at least d + 2 points, there exists a Radon partition for P. When P is in general position in R d and P has exactly d + 2 elements, the partition is unique.