Math Conjectures

Strict Convexity of the First-Passage Percolation Limit Shape

OPENMajorConjectureProposed c. 1965 · Canonical special case

Canonical statement

Assign independent mean-one exponential passage times to the nearest-neighbor edges of Zd\mathbb Z^d, d≥2d\ge2. Let B⊂Rd\mathcal B\subset\mathbb R^d be the deterministic compact convex limit shape from the first-passage percolation shape theorem. Then B\mathcal B is strictly convex: its boundary contains no nontrivial line segment.
View source LaTeX
Assign independent mean-one exponential passage times to the nearest-neighbor edges of \(\mathbb Z^d\), \(d\ge2\). Let \(\mathcal B\subset\mathbb R^d\) be the deterministic compact convex limit shape from the first-passage percolation shape theorem. Then \(\mathcal B\) is strictly convex: its boundary contains no nontrivial line segment.
PROB-004:[HammersleyWelsh1965FPP][AuffingerDamronHanson2017FPP]
Probability theoryGeometry and topology

Notes

The first-passage percolation shape theorem produces a deterministic convex limit body [HammersleyWelsh1965FPP]. Its boundary is expected to be curved for continuous iid weights, yet standard techniques do not prove even the absence of a line segment for exponential edge times. The modern monograph records this as a basic geometric gap [AuffingerDamronHanson2017FPP]. The entry fixes a canonical distribution so it is a precise problem rather than a vague universality claim.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.