Universality of Real Table Polytopes

Definitions and mathematical statement

A real polytope is a nonempty bounded set $P=\operatorname{conv}\{v_1,\ldots,v_N\}\subset{\mathbb{R}}^d$ with real vertices. A line-sum table polytope is formed from an array $x=(x_{i_1,\ldots,i_r})$ of dimensions $m_1\times\cdots\times m_r$, with nonnegative entries and fixed sums along each line. For a chosen collection of real margins $b^{(t)}$, the full line-sum version is \[ T(b)=\left\{x\in{\mathbb{R}}_{\geq0}^{m_1\times\cdots\times m_r}: \sum_{i_t=1}^{m_t}x_{i_1,\ldots,i_r} =b^{(t)}_{i_1,\ldots,i_{t-1},i_{t+1},\ldots,i_r} \text{ for all }t\text{ and remaining indices}\right\}. \] Only consistent margins giving a nonempty polytope are relevant. Real, possibly irrational margins must be allowed if the intended extension is beyond the rational case.

The source asks whether every real polytope can be represented as a table polytope. Under the strong affine-representation interpretation, the target is \[ \forall P\ \exists r,m_1,\ldots,m_r,b\ \exists \text{ affine bijection }F:P\longrightarrow T(b). \] This display makes an interpretation explicit, not a verified choice of the source's meaning of "representable". Affine equivalence, a coordinate projection bijective on the table polytope, representation as a face, arbitrary linear-image representation, and combinatorial equivalence are different requirements. The record does not state which is intended, whether the array order is fixed (for example three-way), or whether prescribed zero entries/faces are allowed. Those distinctions are essential; the displayed general tensor notation must not silently replace a fixed-format universality question.

The supplied literature triage checked 21 August 2026 attributes a rational-polytope universality theorem to De Loera and Onn, while saying that the report leaves the arbitrary-real case open and that its reviewer located no resolution of that case. The exact format and representation strength of the rational theorem are not reproduced here, so it is not promoted to the real assertion or used to settle it by approximation.

Short English statement

Can every polytope with real vertices be realized by nonnegative arrays with prescribed line sums? The supplied assessment distinguishes a known rational universality theorem from the real case. The exact array format and whether representation means affine equivalence, a face or a projection still require source review.

Sources

• Local UnsolvedMath registry, permanent ID 30000112, exact source code catalogue entry; Universality of Real Table Polytopes. Proposed year 2004, status open; corrected_verified question preserves real polytopes and line-sum tables.
• Supplied report citation: Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/39, pp. 2071--2102, https://doi.org/10.4171/owr/2004/39. Title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: "De Loera--Onn", https://epubs.siam.org/doi/abs/10.1137/040610623; "Universality survey", https://math.ucdavis.edu/~deloera/RECENT_WORK/ipcouniversality.pdf; "OWR report", https://doi.org/10.4171/owr/2004/39. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims; linked PDFs were not accessed here.