Three Rings of Polyhedral Simple Functions

Three Rings of Polyhedral Simple Functions

We survey three ways to multiply elements of the additive subgroup of the group of real-valued functions on R(d) which is generated by the indicator functions of polyhedra. In the resulting commutative rings, identities often correspond to useful techniques of decomposition of polyhedra. We are led immediately to various interesting topics, including Ehrhart polynomials, mixed volumes, Gram's r...

Generalized Rings of Measurable and Continuous Functions

This paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

Detecting integral polyhedral functions

We study the class of real-valued functions on convex subsets of Rn which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xia...

Rings of Quotients of Rings of Functions

Strong polyhedral approximation of simple jump sets

We prove a strong approximation result for functions u ∈ W 1,∞(Ω \ J), where J is the union of finitely many Lipschitz graphs satisfying some further technical assumptions. We approximate J by a polyhedral set in such a manner that a regularisation term η(Div u), (i = 0, 1, 2, . . .), is convergent. The boundedness of this regularisation functional itself, introduced in [T. Valkonen: “Transport...