  Let H be the supremum of finitely many real polynomials of degree d and
assume that H has a strict local minimum at 0. We prove a \L ojasiewicz-type
inequality $H(x_1,...,x_n) > ||(x_1,...,x_n)||^s$ where s depends only on d and
n. This implies a similar inequality where $(x_1,...,x_n)$ runs through the
points of a semi-algebraic set.
