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 $1 < p_1 < p_2 \leq \infty$, 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：我们考虑如何计算(双面)Hausdorf 单位 $@p ⁇ 1 ⁇ 1 ⁇ 1 ⁇ 1 美元和$$@ellp ⁇ 2 ⁇ 2 ⁇ 美元标准球之间在一定维度Euclidean空间中1美元 < p_1 < p_2\ p_2\leq\infty$的距离, 并得出一个类似封闭式公式。 我们还为Hausdorf 单位在 $_ 1美元和 $k_ 2美元单位美元- 诺姆球之间计算一个封闭式公式, 这些公式是一定的多面标准球, 以美元为标准值, 1\leq k_1 < k_2\\leq d$。 当两个不同的标准球通过共同线性地图转换时, 我们获得一些关于Hausdordforf 所产生矩形的距离的公式的估计数。 这些估计数是Hausdorff 距离, 取决于线性地图是任意的还是随机的。 然后我们将Hausdforfflexflexf 的两种设定值单位的长度距离的长度长度长度长度球的长度推算, 用来显示一个不同的直线性直线性直线性系统 显示一个直线性的直线性系统, 显示的直线性系统 显示一个直线性直线性直线性直线性直图。