Multi-variable Sinc Integrals and the Volumes of Polyhedra

Using classical Fourier transform techniques we establish inequalities for integrals of the form \[ \int_0^\infty \prod_{k=0}^n \frac{\sin (a_k x)}{a_k x}\,dx. \] These are related to volumes of polyhedra and to quite striking closed form evaluations. (Based in part on a paper by David and Jonathan Borwein to appear in the {\em Ramanujan Journal}; see also Preprint 99:142 at {\sf www.cecm.sfu.ca/preprints}.)