The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $\mu$ in $\mathbb{R}^d$ is defined as the infimum of the $\mu$-masses of all closed halfspaces containing $x$. A natural question is whether the halfspace depth, as a function of $x \in \mathbb{R}^d$, determines the measure $\mu$ completely. In general, it turns out that this is not the case, and it is possible for two different measures to have the same halfspace depth function everywhere in $\mathbb{R}^d$. In this paper we show that despite this negative result, one can still obtain a substantial amount of information on the support and the location of the mass of $\mu$ from its halfspace depth. We illustrate our partial reconstruction procedure in an example of a non-trivial bivariate probability distribution whose atomic part is determined successfully from its halfspace depth.

中文翻译：

---

从半空间深度部分重建测量值

$d$ 维点 $x$ 相对于 $\mathbb{R}^d$ 中有限（或概率）Borel 测度 $\mu$ 的半空间深度定义为 $\mu$ 的下确界- 包含 $x$ 的所有闭合半空间的质量。一个自然的问题是半空间深度是否作为 $x \in \mathbb{R}^d$ 的函数，是否完全决定了度量 $\mu$。一般来说，事实证明并非如此，两种不同的度量有可能在 $\mathbb{R}^d$ 的任何地方都有相同的半空间深度函数。在本文中，我们表明，尽管有这个负面结果，人们仍然可以从其半空间深度获得大量关于支撑和质量的位置的信息。