  Valid inference after model selection is currently a very active area of
research. The polyhedral method, pioneered by Lee, et al. (2016), allows for
valid inference after model selection if the model selection event can be
described by polyhedral constraints. In that reference, the method is
exemplified by constructing two valid confidence intervals when the Lasso
estimator is used to select a model. We here study the length of these
intervals. For one of these confidence intervals, which is easier to compute,
we find that its expected length is always infinite. For the other of these
confidence intervals, whose computation is more demanding, we give a necessary
and sufficient condition for its expected length to be infinite. In
simulations, we find that this sufficient condition is typically satisfied,
unless the selected model includes almost all or almost none of the available
regressors. For the distribution of confidence interval length, we find that
the $\kappa$-quantiles behave like $1/(1-\kappa)$ for $\kappa$ close to $1$.
Our results can also be used to analyze other confidence intervals that are
based on the polyhedral method.
