If d can occur equally likely in the range [0..D] then we have a uniform distribution on the probability of 
with an upper bound of D. So
and we know by figure 3 that the minimum distance traveled occurs for strategy j=d. What strategy is now our best? That follows from
easily enough.
Presumably the more general case is simply found by integrating over
the probability distribution, whatever that may be
. The
general case can be arrived at then by
where 
is the boundary of the enclosing space.