论文标题
关于标准球与线性图之间的Hausdorff距离的注释
A Note on the Hausdorff Distance between Norm Balls and their Linear Maps
论文作者
论文摘要
我们考虑计算单位$ \ ell_ {p_ {p_ {1}} $和$ \ ell_ {p_ {2}} $ norm norm norm norm balls的（$ \ ell_ {p_ {1}} $之间的（双面）hausdorff的距离的问题我们还为$ k_1 $和$ k_2 $ $ d $ d $ norm球之间的Hausdorff距离提供了封闭式公式，它们是$ D $ D $ dimensions in $ 1 \ leq k_1 <k_1 <k_2 <k_2 \ leq d $的某些多面标准球。当两个不同的$ \ ell_p $ norm Ball通过通用线性地图转换时，我们获得了所得凸集之间的Hausdorff距离的几个估计值。这些估计值上限距离豪斯多夫距离或其期望，具体取决于线性图是任意的还是随机的。然后，我们通过将线性地图的参数族族的参数系列在不同的$ \ ell_p $单位标准球上应用，然后以极限的限制性的方式将Minkowski的总和概述，从而概括了两个设置值的积分之间的发展。为了说明应用程序，我们表明，计算具有不同单元标准球值输入不确定性的线性动态系统的hausdorff距离之间的问题，将其减少到此设置值值的积分设置。
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.
