Let K⊂R^n be an origin-symmetric convex body and let Lambda⊂R^n be a lattice. For 1≤ i≤ n, define the i-th successive minimum by lambda_i(K,Lambda)=inflambda>0:dim operatornamespan(lambda K capLambda)≥ i. Then #(K capLambda)≤∏_i=1^n(2 lfloorfrac1lambda_i(K,Lambda)rfloor+1). A nonsymmetric extension, involving the successive minima of a general convex body, is also conjectured and remains open in full generality.

Betke–Henk–Wills lattice-point conjecture

Let $K\subset\mathbb{R}^n$ be an origin-symmetric convex body and let $\Lambda\subset\mathbb{R}^n$ be a lattice. For $1\le i\le n$ , define the $i$ -th successive minimum by $\lambda_i(K,\Lambda)=\inf\{\lambda>0:\dim\operatorname{span}(\lambda K\cap\Lambda)\ge i\}$ . Then

\#(K\cap\Lambda)\le\prod_{i=1}^n\left(2\left\lfloor\frac{1}{\lambda_i(K,\Lambda)}\right\rfloor+1\right).

A nonsymmetric extension, involving the successive minima of a general convex body, is also conjectured and remains open in full generality.

Betke, Henk, and Wills conjectured that the number of lattice points in a symmetric convex body is bounded by a product determined by its successive minima. The conjecture also has a nonsymmetric extension; neither version is known in full generality.

Known results

The symmetric conjecture holds in dimension $2$ (Betke–Henk–Wills).
The nonsymmetric extension holds through dimension $3$ (Malikiosis, 2012).
Tointon’s October 1, 2024 paper gives improved bounds and verifies the conjecture asymptotically in a regime where the relevant successive minimum tends to $0$ .
Wang proves the bound for axis-aligned orthogonal parallelotopes and gives stability results for related shapes (2026).

October 2026 rank-two development

Fei Xue’s preprint establishes a sufficient discrete-monotonicity condition for the Betke–Henk–Wills bound and its nonsymmetric extension in arbitrary ambient dimension when the rank is at most $2$ . This is substantial partial progress, but it does not settle higher-rank cases; the preprint’s claim remains unverified here.

Current status (as of October 2026): The symmetric conjecture is settled in dimension $2$ and the nonsymmetric extension through dimension $3$ , while Xue’s rank-two sufficient condition is claimed in arbitrary dimension; the full conjecture and higher-rank cases remain open.

Betke–Henk–Wills lattice-point conjecture

Progress summary

Solutions 0