Geometric Criteria for Ehrhart Positivity

Definitions and mathematical statement

An integral polytope is a nonempty polytope $P=\operatorname{conv}(v_1,\ldots,v_m)\subseteq{\mathbb{R}}^n$, where $v_j\in{\mathbb{Z}}^n$. Write $d=\dim P$; no full-dimensionality assumption is imposed. Its Ehrhart polynomial is the polynomial $L_P(t)$ characterized by \[ L_P(t)=\#(tP\cap{\mathbb{Z}}^n)\qquad(t\in{\mathbb{Z}}_{\geq 0}), \qquad tP=\{tx:x\in P\}. \] Write \[ L_P(t)=\sum_{i=0}^{d}a_i(P)t^i. \] For this draft, interpret Ehrhart positive in the standard strict sense \[ a_i(P)>0\qquad\text{for every }i\in\{0,\ldots,d\}. \] Thus the condition concerns the coefficients through the actual degree $d$, including the constant coefficient; coefficients beyond degree $d$ are not tested. The supplied record does not explicitly define its positivity convention, so this interpretation requires source confirmation.

Research question. Which geometrically described families $\mathcal C$ of integral polytopes satisfy \[ \forall P\in\mathcal C\;\forall i\in\{0,\ldots,\dim P\},\qquad a_i(P)>0, \] and what geometric criteria, constructions, or arguments can establish this property?

The source specifies neither candidate families nor a precise meaning of geometric methods or a required form of classification. Consequently, this is a research question seeking families and geometric proofs, rather than a single conjecture with fixed hypotheses. The supplied open label and literature assessment are dated curation, not mathematical conclusions verified in this draft.

Short English statement

Determine which families of lattice polytopes have strictly positive coefficients in their Ehrhart polynomials, and develop geometric proofs of that positivity. The intended families and methods are unspecified, and the strict positivity convention requires confirmation from the source.

Sources

• Local UnsolvedMath registry: permanent ID $30003599$, title Geometric Criteria for Ehrhart Positivity, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year 2017 and no named proposer.
• Supplied report citation: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. Supplied source URL: https://doi.org/10.4171/owr/2017/44. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2018 date and the record's 2017 proposed year are retained without resolving their relationship.
• Supplied literature reference: Liu, positivity of Ehrhart polynomials, https://arxiv.org/abs/1711.09962.
• Supplied literature reference: Jochemko--Ravichandran, generalized permutohedra, https://arxiv.org/abs/1909.08448.
• Supplied literature reference: original OWR report, https://doi.org/10.4171/OWR/2017/44.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Ehrhart positivity is established for several families and disproved for others, while no general classification of families of integral polytopes with positive Ehrhart coefficients is available. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record requests independent verification of time-sensitive status and cited claims.