Geometry Encoded by Newton–Okounkov Bodies

Definitions and mathematical statement

The supplied question concerns a pair $(X,D)$ and its Newton--Okounkov bodies, but does not specify the class of varieties, the ground field, the type or positivity of $D$, or the auxiliary valuations or flags.

For notation only, one conventional setting is the following. Let $X$ be an integral projective variety of dimension $n\geq 1$ over an algebraically closed field, let $D$ be a big Cartier divisor, and choose an admissible flag \[ Y_\bullet:\quad X=Y_0\supset Y_1\supset\cdots\supset Y_n=\{p\}, \] where each $Y_i$ is irreducible of codimension $i$ in $X$, and all members of the flag are nonsingular at $p$. For a nonzero section $s\in H^0(X,\mathcal O_X(mD))$, the flag valuation $\nu_{Y_\bullet}(s)\in{\mathbb{Z}}_{\geq 0}^n$ records successive orders of vanishing along the flag: at each stage one removes the previously measured vanishing and restricts to the next member. The associated Newton--Okounkov body is \[ \Delta_{Y_\bullet}(X,D) = \overline{\operatorname{conv}}\left( \bigcup_{\substack{m\in{\mathbb{Z}}_{>0}\\H^0(X,\mathcal O_X(mD))\neq 0}} \left\{\frac{\nu_{Y_\bullet}(s)}{m}:0\neq s\in H^0(X,\mathcal O_X(mD))\right\} \right)\subseteq{\mathbb{R}}^n. \] These hypotheses explain a standard construction; they are not hypotheses supplied by the source and do not resolve its intended scope.

Source question. What geometric data of a pair $(X,D)$ are encoded, and how, by the combinatorics of its Newton--Okounkov bodies?

A precise theorem or conjecture cannot be extracted uniquely from this question as supplied. In particular, the source does not specify which geometric data are sought, what constitutes the combinatorics of a body, or whether one is given a single body, a collection indexed by specified flags or valuations, or all such bodies. No polyhedrality hypothesis is supplied, so an interpretation solely in terms of polytope face lattices would require additional qualification. The intended task is therefore retained as a general research question, with these formulation choices unresolved.

Short English statement

The question asks which geometric features of $(X,D)$ can be recovered from combinatorial information about its Newton--Okounkov bodies, and by what correspondence. The permitted pairs, auxiliary choices, meaning of combinatorics, and desired recovery statements remain unspecified.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30001867$, source problem code catalogue entry, title "Geometry Encoded by Newton--Okounkov Bodies," in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2011.
• Supplied source citation: Mini-Workshop: New Developments in Newton-Okounkov Bodies (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/41, pp. 2327--2363. Original OWR report: https://doi.org/10.4171/owr/2011/41. The citation's parenthetical year 2012 and the record's proposed year 2011 are preserved without reconciliation.
• Supplied literature source: "Newton--Okounkov bodies of flag varieties," https://academic.oup.com/imrn/article/2021/12/9567/5940749.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that modern special-case constructions do not settle the general request for a geometric interpretation of the combinatorics. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification description is supplied provenance, not verification performed for this draft.