Few-vertex Mahler bounds and pyramid cores

Definitions and mathematical statement

For a full-dimensional convex body $K\subset{\mathbb{R}}^n$ and $z\in\operatorname{int}K$, define \[ K^z=\{y:\langle y,x-z\rangle\le1\ \forall x\in K\},\qquad \mathcal P(K)=\min_{z\in\operatorname{int}K}|K|\,|K^z|. \] The minimizer is its Santaló point $s(K)$. Put $M_n=(n+1)^{n+1}/(n!)^2$, the simplex product. The source asks to extend the Meyer--Reisner theorem from polytopes with at most $n+3$ vertices, or at most $n+3$ facets, to a threshold $N(n)>n+3$: every such full-dimensional polytope with at most $N(n)$ vertices (respectively facets) should satisfy \[ \mathcal P(K)\ge M_n, \] with equality exactly for $n$-simplices. The supplied verification explicitly corrects the extracted strict inequality to a non-strict one and restores the powers and factorials. Nonempty interior is retained. The immediate next vertex threshold is $n+4$; no general threshold beyond $n+3$ is proved by the aid.

Its partial reduction concerns pyramids. For a $(d-1)$-dimensional body $B\subset{\mathbb{R}}^{d-1}$ with $s(B)=0$, $d\ge2$, let \[ P=\operatorname{conv}((B,0)\cup\{(0,1)\})\subset{\mathbb{R}}^d. \] The aid reports \[ s(P)=(0,1/(d+1)),\qquad \mathcal P(P)=\frac{(d+1)^{d+1}}{d^{d+2}}\mathcal P(B), \] and the multiplier is exactly $M_d/M_{d-1}$. Hence the normalized ratio is unchanged: \[ \frac{\mathcal P(P)}{M_d}=\frac{\mathcal P(B)}{M_{d-1}}. \] For polytopes, adding a pyramid apex also increases both dimension and vertex count by one. Iteratively removing pyramid apices therefore preserves the excess $\#\text{vertices}-\dim$ and this normalized ratio, reducing a pyramid chain to a nonpyramidal core.

The aid concludes that the $n+4$-vertex pyramid-chain case reduces to cores with four excess vertices. Planar cores satisfy the requested bound unconditionally by the supplied planar result. Three-dimensional cores are said to satisfy it subject to the June 2026 preprint version arXiv:2605.09334v3. Conditional on that preprint's claimed dimension-three theorem, the remaining unresolved cores in this chain reduction have dimension at least four. Without accepting that input, the three-dimensional step remains conditional. The reduction does not cover all nonpyramidal $n+4$-vertex polytopes in arbitrary dimension or supply the requested general improved threshold; equality assertions from any external core theorem also require their full hypotheses.

Short English statement

The source asks to extend the few-vertex or few-facet Mahler bound beyond $n+3$. The supplied aid shows that the normalized volume product is unchanged by pyramiding, reducing that part of the $n+4$-vertex problem to nonpyramidal cores, with the three-dimensional input explicitly conditional on a 2026 preprint.

Sources

• Local UnsolvedMath registry, permanent ID $20001285$, source code catalogue entry; aim-convex-geometry-notes.json, notes[16], tag problem.
• AIM workshop "Mahler's conjecture and duality in convex geometry," S. Reisner's problem, identified by the supplied verification as Problem 21 despite the extracted leading "1": https://aimath.org/WWN/mahlerduality/mahlerduality.pdf. Statement status exact; the verification corrects strictness and superscript loss. No PDF was accessed for this draft.
• M. Meyer and S. Reisner, source theorem [MR], at most $n+3$ vertices or facets; no full bibliographic entry is supplied.
• Dimension-three preprint, supplied as a June 2026 version: https://arxiv.org/abs/2605.09334v3. Its title, authors, exact theorem, and review status are not supplied; the aid's use is conditional. The planar input and pyramid formula have no full authoritative citation in the record.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor label proved, novelty confidence low. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.