  We introduce the notion of consistent error bound functions which provides a
unifying framework for error bounds for multiple convex sets. This framework
goes beyond the classical Lipschitzian and H\"olderian error bounds and
includes logarithmic and entropic error bounds found in the exponential cone.
It also includes the error bounds obtainable under the theory of amenable
cones. Our main result is that the convergence rate of several projection
algorithms for feasibility problems can be expressed explicitly in terms of the
underlying consistent error bound function. Another feature is the usage of
Karamata theory and functions of regular variations which allows us to reason
about convergence rates while bypassing certain complicated expressions.
Finally, applications to conic feasibility problems are given and we show that
a number of algorithms have convergence rates depending explicitly on the
singularity degree of the problem.
