Dimension-Uniform Lower Bounds for Twice-Smooth Integration

Definitions and mathematical statement

Let $d$ be a positive integer and let $D_d\subset{\mathbb{R}}^d$ be a bounded convex domain. The supplied record specifies \[ \overline C^2(D_d)=\left\{f\in C^2(D_d):\ \|f\|_\infty\leq 1,\ \operatorname{Lip}(f)\leq d^{-1/2},\ \operatorname{Lip}(D_\theta f)\leq d^{-1}\right\}. \] Here $C^2(D_d)$ denotes twice continuously differentiable functions, $\|f\|_\infty=\sup_{x\in D_d}|f(x)|$, and $D_\theta f$ denotes differentiation in direction $\theta$. The notation $\operatorname{Lip}$ denotes a Lipschitz seminorm; the record does not explicitly specify its underlying norm or metric. It also does not specify the range or quantifier for $\theta$. In particular, requiring the last inequality for every Euclidean unit direction would be an additional interpretation requiring confirmation.

The proposed conjecture is that there exists an absolute constant $C>0$, independent of dimension and domain, such that \[ e_n\left(\overline C^2(D_d),\operatorname{INT}\right)\geq C n^{-2/d} \] for every positive integer $d$, every admissible $D_d$, and every positive integer $n$. Positive integers are the implicit index ranges used here; the supplied statement does not spell them out. The quantity $e_n$ is described only as the $n$th minimal integration error.

This remains an incompletely specified conjecture: the supplied record does not define $\operatorname{INT}$, its measure or normalization, the information available to algorithms, the meaning of the information budget $n$, or the error criterion. It does not state any volume or geometric normalization of $D_d$, nor any boundary extension requirement on $f$. These choices, together with the directional quantifier, must be recovered from the source before a precise mathematical formulation can be certified; none is imposed here.

Short English statement

The supplied conjecture asks for a lower bound $C n^{-2/d}$ on the minimal integration error for a class of twice continuously differentiable functions with dimension-dependent Lipschitz bounds, using one constant for all dimensions and admissible convex domains. The directional condition, domain normalization, and integration model require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004381$, source problem code catalogue entry, title: Dimension-Uniform Lower Bounds for Twice-Smooth Integration. This attribution refers to the supplied complete registry record.
• Supplied report citation: New Perspectives and Computational Challenges in High Dimensions (2021), Oberwolfach Reports, pp. 377--416, DOI 10.4171/owr/2020/6. Supplied source and literature-review link: https://doi.org/10.4171/owr/2020/6. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No individual proposer is supplied; the proposed year is recorded as 2020.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified here. It labels the problem open and describes the dimension- and domain-uniform lower bound as a stated conjecture, sharp for the cube with grid information; it reports that no general proof or counterexample was found. The supplied background explicitly requests independent verification of time-sensitive status and cited claims. Its listed checked source is the report linked above.
• The registry labels the statement corrected_verified and reports that the official report was checked and the function class and dimension-uniform conjecture restored. These are supplied verification claims, not verification performed for this draft.