The largest lattice body with one interior lattice point
How large can a convex body be when its center is its only interior whole-number point?
Put a convex body on the integer grid. Its balance point is the origin, and the origin is its only interior grid point. In 1964, Eugène Ehrhart predicted the exact largest possible volume. This manuscript claims the sharp bound in every dimension: no such body beats the centered simplex.1manuscript
Volume on an integer grid
A lattice point has integer coordinates. A body is convex when every segment joining two of its points remains inside it; its barycenter is the geometric center of mass.
Assume the barycenter is the origin and that no other lattice point lies strictly inside. Ehrhart asked for the exact volume ceiling in n dimensions. He predicted (n+1)ⁿ/n!, reached by the n-dimensional analogue of a triangle.2manuscript
The hard part is the factorial. Symmetry arguments can control overall width, but they forget how n independent directions combine to make a simplex.
At the sharp centered simplex, the origin is the only interior lattice point.
What was known—and what kept failing
Ehrhart proved the planar case and the all-dimensional simplex case. For arbitrary centered bodies, symmetrizing the body and applying Minkowski’s theorem gave vol(K) ≤ 4ⁿ, far above the sharp simplex value.3manuscript
Thin-shell methods later improved the general estimate to 4ⁿe−c√n. This was a meaningful advance, but it did not recover the exact factorial constant.4Huang
Toric geometry had already proved the sharp inequality for broad structured classes, including certain rational polytopes with controlled facet normals. That route did not cover every convex body—especially an arbitrary irrational one.5Berman
Why harmonic symmetrization stalls
A harmonic symmetral turns K into a centrally symmetric lattice-free body, so Minkowski applies. Remarkably, it has exactly the right volume for the extremal simplex. But the needed comparison depends on the full distribution of simplex coordinates; centroid bounds see only coarse averages and lose the factor n!. The method recognized the right extremizer without explaining its volume.6walkthrough
Stop symmetrizing the body. Convert it into a convex potential, convert lattice points into holomorphic monomials, and make the desired volume appear between a lower and an upper slope of one convex function.
Astra’s route: from a real body to a complex slope
1. Give any centered body a toric potential
A theorem of Berman and Berndtsson supplies a smooth strictly convex function φ on ℝⁿ whose gradient maps ℝⁿ onto the interior of K and satisfies det D²φ = e−φ. Thus the density e−φ has total mass vol(K). Crucially, this works directly for an arbitrary centered convex body; rational vertices are unnecessary.7Berman
2. Turn interior lattice points into monomials
Move to the complex torus (ℂ*)ⁿ and read ordinary real coordinates as logarithms of complex radii. At level k, the square-integrable Laurent monomials zm are indexed exactly by integer points m inside kK. Therefore dim Hk = #(int(kK) ∩ ℤⁿ), and the original “one interior lattice point” assumption becomes H1 = ℂ: only the constant monomial survives.8manuscript
3. Jet counting forces a lower slope
At p = (1,…,1), filter Hk by order of vanishing. Vanishing to order j kills every Taylor coefficient of degree below j, at most (n+j−1 choose n) conditions. Summing these dimensions and passing to the lattice-counting limit yields an initial-slope lower bound containing (n! vol(K))1/n. This is where the missing n! finally appears: it is the leading coefficient of n-variable Taylor-jet counting.9manuscript
4. Level one turns Bergman positivity into ordinary convexity
The filtered bases define a limiting ray of complex potentials ψt and a normalized partition function L(t). Simply integrating pointwise-convex rays would be invalid: a variance term can have the wrong sign. The walkthrough explicitly identifies this failed shortcut.10walkthrough
The rescue is H1 = ℂ. Because the relevant holomorphic space has rank one, its Bergman kernel is exactly the inverse partition function. Berndtsson’s positivity theorem then makes L genuinely convex—not merely pointwise plausible.11Berndtsson12manuscript
5. A shrinking complex ball forces the upper slope
A section that vanishes to high order becomes tiny near p. In the filtered ray, the compensating neighborhood has complex radius proportional to e−t/2. Since ℂⁿ has real dimension 2n, its volume scales like e−nt. This gives L(t) ≤ nt + O(1), hence convexity forces L′+(0) ≤ n.13manuscript
The factorial and the exponent meet in one squeeze
The walkthrough’s central insight is that the lower constant comes from an n-dimensional simplex of Taylor multi-indices, while the upper n comes from the 2n real dimensions of a complex ball.14walkthrough
Technical layer · the slope calculation
Set V = vol(K) and cK = (n!V)1/n. The jet filtration Fkj has codimension at most (n+j−1 choose n). After truncating vanishing orders at cKk, a Riemann-sum limit gives ∫g dμ ≥ [n/(n+1)]cK. Because L′+(0) = ∫g dμ, this is the lower slope.15manuscript
The potential obeys ∇φ(ℝⁿ) = int(K) and det D²φ = e−φ. Weighted integrability gives Hk = span{zm : m ∈ int(kK)∩ℤⁿ}, while lattice counting gives dim Hk/kⁿ → V. The rank-one level H1 = ℂ converts Berndtsson positivity into convexity of L; the local ball estimate gives its upper slope. Hence [n/(n+1)](n!V)1/n ≤ n.16manuscript
Exact theorem, equality, and limits
Theorem 1.1. Let K ⊂ ℝⁿ be full-dimensional, compact, and convex, with barycenter 0. If int(K)∩ℤⁿ = {0}, then vol(K) ≤ (n+1)ⁿ/n!. The centered simplex (n+1)Δn − (1,…,1) satisfies the hypotheses and reaches equality.17manuscript
The theorem establishes the best numerical constant. It does not classify all bodies attaining equality. The expected uniqueness statement—that every extremizer is a unimodular image of the centered simplex—remains a separate refinement.18Nill
The public EhrhartVolumeInequality.lean file contains a formal set-level theorem with convexity, compactness, nonempty interior, barycenter, and unique-interior-lattice-point hypotheses leading to the sharp bound. This is substantial machine-checked evidence for the encoded theorem, but still warrants independent review of the definitions and analytic interfaces.19formal artifact
Is the claim overhyped?
If the manuscript and formalization withstand expert scrutiny, this is a major resolution. The manuscript settles the numerical inequality; the equality classification and independent validation remain open.
“Ehrhart’s volume conjecture is proved.”
At manuscript level, yes: Theorem 1.1 is exactly the sharp all-dimensional numerical inequality with an equality example.20manuscript
“The extremal body is completely understood.”
No. The centered simplex is an extremizer, but the manuscript explicitly does not classify every equality case.21manuscript
“Astra solved it.”
That is OpenAI’s attribution. Its announcement, manuscript, walkthrough, and Lean file are same-release evidence—not an independent referee report or reproduction.22announcement
Full bibliography
22 fully annotated sources
- 01 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Chapter 8, abstract and Theorem 1.1, pp. 217–218. The released manuscript and source of the main theorem; not independent validation. ↩
- 02 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Introduction, pp. 217–218. Problem statement, sharp simplex, and historical context. ↩
- 03 · primary manuscript
OpenAI. Previous work on Ehrhart’s conjecture — p. 218, “Previous work”. Locates Ehrhart’s planar and simplex cases and the general 4^n symmetrization bound. ↩
- 04 · peer-reviewed predecessor
Han Huang, Boaz Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou. Improved bounds for Hadwiger’s covering problem via thin-shell estimates — JEMS 24 (2022), Proposition 6.2. A pre-resolution general bound of 4^n exp(−c√n). ↩
- 05 · peer-reviewed predecessor
Robert Berman and Bo Berndtsson. The volume of Kähler–Einstein Fano varieties and convex bodies — Corollary 1.4 and Theorem 1.5. Proved the sharp inequality for broad structured classes of convex bodies and rational polytopes. ↩
- 06 · reasoning walkthrough
OpenAI. Reasoning Walkthroughs: The Sharp Ehrhart Inequality — Chapter 9, §9.2, pp. 36–37. Retrospective account of harmonic symmetrization and why it lost the factorial; not independent evidence. ↩
- 07 · peer-reviewed predecessor
Robert Berman and Bo Berndtsson. Real Monge–Ampère equations and Kähler–Ricci solitons on toric log Fano varieties — Theorem 1.1. Existence of the convex transport potential for an arbitrary centered body. ↩
- 08 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Lemma 2.1, pp. 219–220. Identifies weighted Laurent monomials with interior lattice points of kK. ↩
- 09 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Lemma 3.1 and Proposition 3.2, pp. 222–224. Jet counting and the sharp lower bound on the initial slope. ↩
- 10 · reasoning walkthrough
OpenAI. Reasoning Walkthroughs: The Sharp Ehrhart Inequality — Chapter 9, §§9.5–9.6, pp. 38–39. Explains the limiting Bergman ray and the finite-level convexity pitfall. ↩
- 11 · peer-reviewed predecessor
Bo Berndtsson. Subharmonicity properties of the Bergman kernel — Theorem 1.1; Annales de l’Institut Fourier 56 (2006). Positivity theorem used to obtain convexity of the rank-one partition function. ↩
- 12 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Lemma 4.1, pp. 224–225. Uses H1 = C to identify the Bergman kernel with the inverse partition and prove convexity. ↩
- 13 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Lemma 4.2, p. 225. Shrinking complex-ball estimate giving the upper slope L′+(0) ≤ n. ↩
- 14 · reasoning walkthrough
OpenAI. Reasoning Walkthroughs: The Sharp Ehrhart Inequality — Chapter 9, §§9.3–9.7, pp. 37–40. Discovery-level proof map from jet counting through the slope squeeze. ↩
- 15 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Equation (2) and §§2–4, pp. 218–225. Technical lower- and upper-slope inequalities. ↩
- 16 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Equations (3)–(8), pp. 219–220. Monge–Ampère normalization and lattice Bergman spaces. ↩
- 17 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Theorem 1.1, p. 218. Exact hypotheses, bound, and sharp centered simplex. ↩
- 18 · peer-reviewed predecessor
Benjamin Nill and Andreas Paffenholz. On the equality case in Ehrhart’s volume conjecture — Conjecture 1.1 and Theorem 1.4. Equality context for structured classes; the global equality classification remains separate. ↩
- 19 · formal certificate
OpenAI. EhrhartVolumeInequality.lean — lines 55737–55755. Lean declaration of the set-level sharp volume inequality under convexity, compactness, full-dimensionality, barycenter, and lattice hypotheses. ↩
- 20 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Theorem 1.1. Supports the numerical inequality and sharpness claim at manuscript level. ↩
- 21 · primary manuscript
OpenAI. The Sharp Inequality in Ehrhart’s Volume Conjecture — Introduction, equality discussion. Explicitly says the manuscript does not classify all equality cases. ↩
- 22 · official announcement
OpenAI. Ten advances in mathematics — item 8. Evidence for OpenAI’s attribution and framing only; not independent mathematical review. ↩