Maximum Simplicial Depth of Point Configurations

Definitions and mathematical statement

For positive integers $d$ and $n\ge d+1$, let $T\subset{\mathbb{R}}^d$ be a set of $n$ distinct points. Interpret general position as requiring every subset of $T$ with at most $d+1$ elements to be affinely independent. For $S\subseteq T$ with $|S|=d+1$, its convex hull $\operatorname{conv}(S)$ is a $d$-simplex. For any $x\in{\mathbb{R}}^d$, define \[ D_T(x)=\#\left\{S\subseteq T:|S|=d+1,\ x\in\operatorname{int}_{{\mathbb{R}}^d}(\operatorname{conv}(S))\right\}. \] Thus simplices are counted by their vertex subsets, and a simplex containing $x$ only on its boundary is excluded. No general-position condition on $T\cup\{x\}$ is imposed.

The extremal question is to determine, as a function of $n$ and $d$, \[ M_d(n)=\max_{\substack{T\subset{\mathbb{R}}^d,\ |T|=n\\ T\text{ in general position}}}\ \max_{x\in{\mathbb{R}}^d}D_T(x). \] The parameter range above is the nontrivial positive-dimensional range; the source does not state parameter bounds or define general position explicitly. The displayed formulation interprets "greatest possible" as allowing both $T$ and $x$ to vary. If $T$ is instead intended to be fixed, the corresponding target is $\max_{x\in{\mathbb{R}}^d}D_T(x)$, which depends on $T$. The supplied background also mentions an abstract version, but gives no definition from which to formulate it.

Short English statement

Determine the largest possible number of simplices with $d+1$ vertices from an $n$-point configuration in general position in ${\mathbb{R}}^d$ whose interiors contain one common point. Boundary containment does not count. The meanings of general position and the optimization over configurations require confirmation against the source.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30001662$, title "Maximum Simplicial Depth of Point Configurations," exact source code catalogue entry. No individual proposer is supplied; the recorded proposed year is 2011.
• Supplied report citation: Combinatorics (2011), Oberwolfach Reports, pp. 5--83, DOI 10.4171/owr/2011/01. https://doi.org/10.4171/owr/2011/01. The registry labels the statement corrected_verified and reports that the official report was checked and the statement repaired to match its passage. This is supplied verification metadata, not an independent verification here.
• Supplied literature reference: Leader--Tan, https://arxiv.org/abs/0906.4027.
• Supplied literature reference: Matoušek--Wagner, https://arxiv.org/abs/1102.3515.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the geometric and abstract versions have strong asymptotic bounds and settled low-dimensional cases, while the exact higher-dimensional extremal value is not known. This assessment is supplied dated curation derived from the OWR material, not independently verified; its status and cited claims require independent checking.