Polynomial Algorithms for the Linear Feasibility Problem

We consider the homogenized linear feasibility problem, to find an x on the unit sphere, satisfying n linear inequalities ai x ≥ 0. To solve this problem we consider the centers of the insphere of spherical simplices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algorithm for the linear feasibility problem. If we allow rescaling this algorithm becomes polynomial. We point out that the algorithm solves as well the more general convex feasibility problem. Moreover numerical experiments show that the algorithm could be of practical interest.