Multi-variable sinc integrals and volumes of polyhedra (Q1611258)

\textit{D. Borwein} and \textit{J. M. Borwein} [Ramanujan J. 5, No. 1, 73-89 (2001; Zbl 0991.42004)] obtained interesting results for evaluating the integrals \[ I_n:=\int_0^\infty \prod_{j=1}^{1+n} \text{ sinc}(a_jx) dx, \] where \(\text{sinc}(x):={\sin x\over x}\). In this paper the authors investigate the multi-variable sinc integrals \[ \sigma(S):=\int_{{\mathbb R}^m} \prod_{k=1}^{m+n} \text{ sinc}(s_ky) dy \] where \(S=(s_1,\cdots,s_{m+n}), \;s_k\in {\mathbb R}^m.\) They obtain the results concerning the relationship between the integral \(\sigma(S)\) and the volume of the associated symmetric convex polyhedron. Also they establish an explicit algebraic formula for the computation of \(\sigma(S)\) which generalizes the result obtained by Borwein and Borwein in the one dimensional case (\(m=1\)) in the paper mentioned above .