Tverberg Depth Bounds from Tukey Depth

Definitions and mathematical statement

Let $d\in{\mathbb{Z}}$ satisfy $d\geq2$, let $S\subset{\mathbb{R}}^d$ be finite, and let $q\in{\mathbb{R}}^d$. The supplied record does not define the two depth functions. For this draft, adopt the following unnormalized counting conventions, subject to source verification. Define the Tukey depth by \[ \operatorname{TD}(S,q)=\min\left\{|S\cap H|:H\subset{\mathbb{R}}^d\text{ is a closed halfspace and }q\in H\right\}. \] Thus points on the boundary of a halfspace are counted. Define the Tverberg depth by \[ \operatorname{TvD}(S,q)=\max\left(\{0\}\cup\left\{k\in{\mathbb{Z}}_{\geq1}:\text{there exist pairwise disjoint nonempty subsets }S_1,\ldots,S_k\subseteq S\text{ with }q\in\bigcap_{i=1}^k\operatorname{conv}(S_i)\right\}\right), \] where $\operatorname{conv}(A)$ denotes the convex hull of $A$. In particular, both depths are zero when $S=\varnothing$.

With these conventions, the supplied conjectural assertion is \[ \forall d\in{\mathbb{Z}}_{\geq2}\;\forall S\subset{\mathbb{R}}^d\text{ finite}\;\forall q\in{\mathbb{R}}^d,\qquad \operatorname{TvD}(S,q)\geq\frac{\operatorname{TD}(S,q)}{d-1}. \] The record imposes no general-position assumption, does not exclude $q\in S$, and supplies no exceptional boundary cases. These quantifiers are retained; the assertion is recorded as a conjecture, not as an established inequality.

The record additionally asks whether this bound is tight, without specifying the meaning of tightness. If this means that, for every fixed dimension, $1/(d-1)$ is the largest coefficient valid uniformly over all finite point sets and query points under the conventions above, the corresponding sharpness assertion would be \[ \forall d\in{\mathbb{Z}}_{\geq2}\;\forall\varepsilon>0\;\exists S\subset{\mathbb{R}}^d\text{ finite}\;\exists q\in{\mathbb{R}}^d:\quad \operatorname{TD}(S,q)>0,\qquad \operatorname{TvD}(S,q)<\left(\frac1{d-1}+\varepsilon\right)\operatorname{TD}(S,q). \] This is an explicitly conditional interpretation of tightness. Whether the source instead intends exact equality examples, an integer-rounded bound, or another asymptotic interpretation remains unresolved. The depth conventions and the compatibility of the stated factor with the full supplied scope require human mathematical review.

Short English statement

The supplied problem asks whether every finite point set in ${\mathbb{R}}^d$, for every integer $d\geq2$ and every query point, has Tverberg depth at least its Tukey depth divided by $d-1$, and whether this factor is sharp. This draft uses explicitly stated counting conventions; the source record leaves those conventions and the meaning of sharpness unspecified.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004515$, source problem code catalogue entry, title "Tverberg Depth Bounds from Tukey Depth." Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Discrete Geometry (2021), Oberwolfach Reports, pp. 1469--1528, DOI 10.4171/owr/2020/30. https://doi.org/10.4171/owr/2020/30. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports; no problem-specific page or passage is supplied.
• Supplied literature source: Schnider, Enclosing Depth and Other Depth Measures. https://link.springer.com/article/10.1007/s00493-023-00045-4.
• Supplied literature source: Oberwolfach Reports 2020/30. https://ems.press/content/serial-article-files/46867.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that the general bound $\operatorname{TvD}(S,q)\geq\operatorname{TD}(S,q)/d$ is known, and describes the factor $1/(d-1)$ as unsettled. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.