Coefficients in Signed Simplex Decompositions

Definitions and mathematical statement

The supplied record refers to a set $P$, its uniform measure $\mu_P$, and coefficients $\alpha_T$ indexed by simplices $T$ in a signed simplex decomposition. It does not specify the admissible sets $P$, the ambient dimension, the normalization of the measures, or the construction that determines the simplices and coefficients.

To express the intended question schematically, let $\mathcal D$ denote the unspecified class of admissible pairs consisting of a set $P$ and a signed simplex decomposition of its uniform measure. For $D\in\mathcal D$, write $\mathcal T_D$ for its simplex index set and $\alpha_{D,T}$ for its coefficients. These symbols are placeholders for the missing source definitions, rather than a specification of a new class of decompositions.

The intended question is whether \[ \forall D\in\mathcal D\;\forall T\in\mathcal T_D, \qquad \alpha_{D,T}\ne 0 \ \Longrightarrow\ \alpha_{D,T}\in\{-1,1\}. \] If this assertion fails, characterize the set of possible nonzero coefficients \[ \left\{\alpha_{D,T}:D\in\mathcal D,\ T\in\mathcal T_D,\ \alpha_{D,T}\ne0\right\}. \] The quantification over $\mathcal D$ must be resolved from the original mathematical context: the supplied record does not establish whether the question concerns one prescribed decomposition for each $P$, a restricted family of decompositions, or all decompositions satisfying some unstated conditions. Consequently, this is a conditional statement template requiring the missing hypotheses before it becomes a fully specified mathematical problem.

Short English statement

In the intended signed simplex decomposition of the uniform measure on $P$, must every nonzero simplex coefficient equal $1$ or $-1$? If not, which nonzero coefficients can occur? The admissible sets, measure normalization, and decomposition rule are missing from the supplied record and require review.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002299$, exact source problem code catalogue entry, title Coefficients in Signed Simplex Decompositions, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2013 and names no proposer.
• Structured Function Systems and Applications (2013), Oberwolfach Reports, Report 11/2013, pp. 579--655, DOI 10.4171/owr/2013/11. Supplied source and literature link: https://doi.org/10.4171/owr/2013/11. The supplied curation identifies the extracted problem as the report's Question 2. The report is described as documenting a mathematical workshop and its open-problem discussions.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports that the cited report gives no proof or counterexample and that no later resolution was found in the checked literature. The listed source checked is Oberwolfach Report 11/2013, at the DOI link above. This is supplied dated curation, not an independently verified literature assessment.
• The registry marks the statement corrected_verified and states that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification claim; the report has not been independently consulted for this draft.