Lattice Embedding of Direction-Preserving Correspondence over Integrally Convex Set

We consider the relationship of two fixed point theorems for direction-preserving discrete correspondences. We show that, for space of no more than three dimensions, the fixed point theorem [5] of Iimura, Murota and Tamura, on integrally convex sets can be derived from Chen and Deng’s fixed point theorem [1] on lattices by expanding every direction-preserving discrete correspondence over an integrally convex set to one over a lattice. We present a counter example for four dimensional space. Related algorithmic results are also presented for finding a fixed point of direction-preserving correspondences on integrally convex sets, for spaces of all dimensions.