Exact Dimensions of Trivariate Spline Spaces

Definitions and mathematical statement

The supplied background concerns trivariate $C^1$ spline spaces on bipyramid cells. Here a trivariate $C^1$ spline means a piecewise polynomial function of three variables whose pieces join with continuous first partial derivatives across their interfaces. The record does not specify the polynomial degree convention, the subdivision of the bipyramid cells, the admissible geometry, the coefficient field, or any boundary conditions.

To display the intended question without supplying these missing hypotheses, let $\Theta_{\mathrm{rem}}$ denote the unspecified family of remaining cases referred to in the source. For $\theta\in\Theta_{\mathrm{rem}}$, write $S_\theta$ for the intended spline space, $K_\theta$ for its intended coefficient field, and $L_\theta,U_\theta$ for the lower and upper dimension bounds referred to in the source, respectively. These symbols are placeholders for source data absent from the supplied record. The question has the schematic form \[ \text{For every }\theta\in\Theta_{\mathrm{rem}}\text{ with } L_\theta\leq\dim_{K_\theta}S_\theta\leq U_\theta \quad\text{and}\quad L_\theta<U_\theta, \qquad \dim_{K_\theta}S_\theta\stackrel{?}{=}U_\theta. \] The supplied statement is: In the remaining cases where the upper and lower dimension bounds differ, does the dimension equal the upper bound?

Neither the bounds nor the family of remaining cases is identified in the supplied record. Thus this is a faithful schematic rendering of the question, not a fully specified standalone conjecture. In particular, it does not quantify over arbitrary spline spaces or arbitrary coefficient fields.

Short English statement

For the trivariate $C^1$ spline spaces on bipyramid cells intended by the source, must the dimension attain the stated upper bound whenever the stated lower and upper bounds differ? The record omits the bounds and the precise remaining cases, so a fully specified formulation requires source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30002869$, title Exact Dimensions of Trivariate Spline Spaces, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2015 and names no proposer. It labels the statement exact and says that it was retained byte-for-byte; these are supplied registry qualifications.
• Supplied source citation: Multivariate Splines and Algebraic Geometry (2016), Oberwolfach Reports, pp. 1139--1200, DOI https://doi.org/10.4171/owr/2015/21. This is also the literature-source link labelled "Oberwolfach Report 21." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No specific page or contribution containing this question is supplied.
• Supplied literature source labelled "Colvin--DiMatteo--Sorokina": https://doi.org/10.1016/j.cagd.2015.12.001. No article title or precise supporting location is supplied.
• Supplied literature source labelled "DiPasquale--Villamizar": https://doi.org/10.1137/20M1341118. No article title or precise supporting location is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the remaining unequal-bound cases for trivariate $C^1$ splines on bipyramid cells are conjectural, and that later lower-bound work does not establish the proposed equality in all cases. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.