  We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp
460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate
polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x =
(x_1,...,x_n)$ on the closed ball ${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$,
computable by geometric programming, for any real $M$. We compare this bound
with the (global) lower bound $f_{{\rm gp}}$ obtained by Ghasemi and Marshall,
and also with the hierarchy of lower bounds, computable by semidefinite
programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our
computations show that the bound $f_{{\rm gp},M}$ improves on the bound
$f_{{\rm gp}}$ and that the computation of $f_{{\rm gp},M}$, like that of
$f_{{\rm gp}}$, can be carried out quickly and easily for polynomials having of
large number of variables and/or large degree, assuming a reasonable sparsity
of coefficients, cases where the corresponding computation using semidefinite
programming breaks down.
