  We present an O(mn^2) algorithm for linear programming over the real numbers
with n primal and m dual variables through deciding the support set a of an
optimal solution. Let z and e be two 2(n+m)-tuples with z representing the
primal, dual and slack variables of linear programming, and e the all-one
vector. Let Z denote the region including all (tz, t) with z meeting the zero
duality gap constraint, all primal and dual constraints except for the
non-negativity constraints, and without limit on the real number t. Let L be
the projection of Z on the hyperplane defined by t = 0. Consider a squeeze
mapping involving the two variables of each complementary pair of z. The
projection of e on the image of L of the mapping lies in an (n+m-1)-sphere Q
centered at e/2 of a diameter whose square equals 2(n+m). The sum of the two
components of a complementary pair of z in Q equals one, and Q is the
circumsphere of the hypercube where each component of its vertices takes value
in {0, 1}. One vertex v* called the solution vertex is the indicator vector of
a. The algorithm uses squeeze mapping to move the aforementioned projection
around v* along Q so that a is identified at certain position. It consists of
O(n) unidimensional squeeze mappings, each of which uses O(mn) arithmetic
operations.
