  We consider the problem of computing the (two-sided) Hausdorff distance
between the unit $\ell_{p_{1}}$ and $\ell_{p_{2}}$ norm balls in finite
dimensional Euclidean space for $1 \leq p_1 < p_2 \leq \infty$, and derive a
closed-form formula for the same. We also derive a closed-form formula for the
Hausdorff distance between the $k_1$ and $k_2$ unit $D$-norm balls, which are
certain polyhedral norm balls in $d$ dimensions for $1 \leq k_1 < k_2 \leq d$.
When two different $\ell_p$ norm balls are transformed via a common linear map,
we obtain several estimates for the Hausdorff distance between the resulting
convex sets. These estimates upper bound the Hausdorff distance or its
expectation, depending on whether the linear map is arbitrary or random. We
then generalize the developments for the Hausdorff distance between two
set-valued integrals obtained by applying a parametric family of linear maps to
different $\ell_p$ unit norm balls, and then taking the Minkowski sums of the
resulting sets in a limiting sense. To illustrate an application, we show that
the problem of computing the Hausdorff distance between the reach sets of a
linear dynamical system with different unit norm ball-valued input
uncertainties, reduces to this set-valued integral setting.
