A piecewise-constant valuation is a kind of a function that represents the utility of an agent over a continuous resource, such as land.
Piecewise-constant and piecewise-uniform valuations are particularly useful in algorithms for fair cake-cutting.
The value-density function assigns, to each point of the resource, a real value.
In other words: there is a partition of the resource C into finitely many regions, C1,...,Ck, such that for each j in 1,...,k, the function v inside Cj equals some constant Uj.
A piecewise-linear valuation is a generalization of piecewise-constant valuation in which the value-density in each region j is a linear function, ajx+bj (piecewise-constant corresponds to the special case in which aj=0 for all j).