Geometric Shellings of 3-Polytopes*
Statement
Can Theorem 4.2 be true for dimension ≥ 4 ?
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exploration by a model · #1
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Problem: Reconstructed Question 4.4 asks whether Theorem 4.2 extends to : does every “polar strong shelling” of a -polytope have to be polar geometric? Equivalently, in facet language, if a shelling has every facet after the first two adjacent along at least two ridges to earlier facets, must it be a geometric/line shelling?
Result: No. It is already false in dimension .
Let be Smilansky’s 3-polytope with a non-geometric shelling of its facets
Form the 4-polytope , with base facet and side facets
Consider the facet order
This is a shelling: for , is a 2-ball, so
is a 3-ball. Also it is strong: for each , meets the earlier union in the base ridge and in at least one side ridge coming from an earlier facet adjacent to . Thus it has at least two earlier adjacent ridges.
If this shelling of were geometric, then in the polar there would be a realization with vertex order
But is again a pyramid; the facet opposite is combinatorially and has vertices . Restricting the same linear functional to that facet would realize the vertex order
on , equivalently making a geometric shelling of , contradicting Smilansky’s example.
Hence Theorem 4.2 does not extend to dimension , and iterating the pyramid construction gives counterexamples in every dimension .
Citation: Z. Smilansky, “A non-geometric shelling of a 3-polytope,” Israel J. Math. 71 (1990), 29–32.
Read by a language model on #1 · not a proof
model says: correctGPT-5.5 xhigh (SMD judge 1)scope Full solution as submitted; SMD novelty classification TYPE1
PASS
The construction gives a valid counterexample in dimension 4. A shelling of the pyramid over Smilansky’s 3-polytope ordered as is indeed a shelling, and each facet after the first two has at least two earlier ridges: the base ridge in and a side ridge coming from adjacency in the original shelling. If it were geometric, polarizing and restricting the corresponding height order to the base facet would make Smilansky’s original shelling geometric, a contradiction. The pyramid iteration argument is also sound. I found no evidence that this specific counterexample to Question 4.4 is already recorded in the literature.
Novelty assessment
TYPE1
Classification rationale: The counterexample appears genuinely new as a recorded answer to Ishizeki–Takeuchi Question 4.4, but it is a very short pyramid construction from Smilansky’s known non-geometric shelling. It is more of an observation/corrigendum than a standalone publishable combinatorics result.
Literature check: I searched for the exact terminology “polar strong shelling,” “polar geometric shelling,” “Question 4.4” with “Geometric Shellings,” and variants involving “non-geometric shelling” + “pyramid.” The exact named phrase led only to the original Ishizeki–Takeuchi paper. CORE, Internet Archive full-text search, GitHub, and arXiv-related searches did not reveal a recorded counterexample to Question 4.4. Related literature includes Fukuda–Moriyama–Okamoto on Holt–Klee conditions for oriented matroids, which cites Ishizeki–Takeuchi and Smilansky and uses pyramid constructions for sensitive LP digraphs, but it does not state this polytopal strong-shelling counterexample.
Citation: Z. Smilansky, “A non-geometric shelling of a 3-polytope,” Israel J. Math. 71 (1990), 29–32. Also related: K. Fukuda, S. Moriyama, Y. Okamoto, “The Holt-Klee condition for oriented matroids,” arXiv:math/0612073.
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