Hirzebruch-Riemann-Roch theorem, convex chains and toric varieties

A convex chain is a finite linear combination of indicator functions of convex polytopes. We review the algebra of convex chains and the correspondence with multi-valued piecewise linear functions. Then, we associate torus equivariant vector bundles on toric varieties to convex chains and show Khovanskii-Pukhlikov Riemann-Roch theorem for convex chains extends the classical Hirzebruch-Riemann-Roch theorem for torus equivariant vector bundles.