Equality Cases in the Balanced Lower Bound Conjecture

Definitions and mathematical statement

Let $P$ be a convex simplicial polytope of dimension $d$, and let $\Delta=\partial P$ be its boundary simplicial complex. We use the convention $d\geq 1$; the supplied record does not specify a lower-dimensional convention. The polytope is balanced if there is a coloring $c:V(\Delta)\to\{1,\ldots,d\}$ such that the endpoints of every edge have different colors. Equivalently, every facet of $\Delta$ contains exactly one vertex of each color.

For $-1\leq j\leq d-1$, let $f_j(\Delta)$ be the number of $j$-dimensional faces, with $f_{-1}(\Delta)=1$. Define the $h$-numbers $h_0(P),\ldots,h_d(P)$ by \[ \sum_{i=0}^{d}h_i(P)t^{d-i} = \sum_{i=0}^{d}f_{i-1}(\Delta)(t-1)^{d-i}. \] The inequality assertion in the supplied record is that every balanced simplicial $d$-polytope satisfies \[ \frac{h_{r-1}(P)}{\binom{d}{r-1}} \leq \frac{h_r(P)}{\binom{d}{r}} \qquad \text{for every integer }1\leq r\leq\left\lfloor\frac d2\right\rfloor. \] This index range is empty when $d=1$.

The remaining proposed target concerns equality \[ \frac{h_{r-1}(P)}{\binom{d}{r-1}} = \frac{h_r(P)}{\binom{d}{r}}. \] The source says that equality at the relevant adjacent terms should characterize the "balanced stacked property." It does not define that property, specify which indices are relevant, or state any index-dependent version or additional dimension restrictions. Consequently, a precise necessary-and-sufficient equality characterization cannot be extracted from this record without further source clarification. No particular definition of balanced stackedness or equality equivalence is imposed here.

The supplied dated curation attributes a proof of the inequality assertion to Juhnke--Kubitzke and Murai and describes the equality characterization as unresolved in the extracted context. These are supplied literature assessments, not independently verified claims.

Short English statement

For a balanced simplicial polytope, the supplied assertion says that its $h$-numbers, divided by the corresponding binomial coefficients, increase weakly through the middle index. The intended remaining question is to characterize equality using a balanced stacked property, but the record omits the definition and the relevant index restrictions. Its supplied status is partially solved.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30002775$, title: Equality Cases in the Balanced Lower Bound Conjecture; exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems; proposed year recorded as 2015. No individual proposer is supplied.
• Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5. Supplied report reference: https://doi.org/10.4171/owr/2015/5. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Juhnke--Kubitzke--Murai, primary paper": https://arxiv.org/abs/1503.06430. The supplied assessment credits Juhnke--Kubitzke and Murai with proving the inequality part of the balanced generalized lower bound conjecture.
• Supplied literature source labeled "Murai, Oberwolfach Report": https://doi.org/10.4171/owr/2015/5.
• Literature checked at: 2026-08-21. This exact date records supplied dated curation, not independent verification. That curation labels the problem partially solved and says that the equality characterization remains unresolved in the extracted context; the background explicitly calls for independent verification of time-sensitive status and cited claims.