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.

翻译：本文考虑有限维欧氏空间中单位ℓ_{p₁}与ℓ_{p₂}范数球在1 ≤ p₁ < p₂ ≤ ∞条件下的（双向）Hausdorff距离计算问题，并推导出其闭式解。进一步地，我们针对1 ≤ k₁ < k₂ ≤ d维空间中的特定多面体范数球——即k₁与k₂单位D-范数球，给出了其Hausdorff距离的闭式表达式。当两个不同的ℓₚ范数球通过同一线性映射变换时，我们获得了所得凸集之间Hausdorff距离的若干估计值。根据线性映射是确定性的还是随机的，这些估计值分别给出了Hausdorff距离或其期望的上界。随后，我们将前述结论推广至一类集值积分之间的Hausdorff距离问题：该问题通过将参数化线性映射族作用于不同ℓₚ单位范数球，再以极限意义下对所得集合进行闵可夫斯基和得到。为说明应用，我们证明：具有不同单位范数球值输入不确定性的线性动态系统可达集之间的Hausdorff距离计算问题，可归约为此类集值积分框架。