  A worst case distribution is a minimiser of the expectation of some random
payoff within a family of plausible risk factor distributions. The plausibility
of a risk factor distribution is quantified by a convex integral functional.
This includes the special cases of relative entropy, Bregman distance, and
$f$-divergence. An ($\epsilon$-$\gamma$)-almost worst case distribution is a
risk factor distribution which violates the plausibility constraint at most by
the amount $\gamma$ and for which the expected payoff is not better than the
worst case by more than $\epsilon$. From a practical point of view the
localisation of almost worst case distributions may be useful for efficient
hedging against them. We prove that the densities of almost worst case
distributions cluster in the Bregman neighbourhood of a specified function,
interpreted as worst case localiser. In regular cases, it coincides with the
worst case density, but when the latter does not exist, the worst case
localiser is perhaps not even a density. We also discuss the calculation of the
worst case localiser, and its dependence on the threshold in the plausibility
constraint.
