  We consider the problem of constructing honest and adaptive confidence sets
in Lp-loss (with p>=1 and p < infinity) over sets of Sobolev-type classes, in
the setting of non-parametric Gaussian regression. The objective is to adapt
the diameter of the confidence sets with respect to the smoothness degree of
the underlying function, while ensuring that the true function lies in the
confidence interval with high probability.When p >=2, we identify two main
regimes, (i) one where adaptation is possible without any restrictions on the
model, and (ii) one where critical regions have to be removed. We also prove by
a matching lower bound that the size of the regions that we remove can not be
chosen significantly smaller. These regimes are shown to depend in a
qualitative way on the index p, and a continuous transition from p = 2 to p =
infinity is exhibited.
