A Weakened Wills Inequality for Convex Bodies

Definitions and mathematical statement

Let $n\geq2$, and let a convex body $K\subset{\mathbb{R}}^n$ mean a compact convex set with nonempty interior. Write \[ G(K)=|K\cap{\mathbb{Z}}^n| \] for its lattice-point enumerator. Use Euclidean intrinsic volumes $V_0(K),\ldots,V_n(K)$, normalized by the Steiner formula \[ \operatorname{vol}_n(K+tB^n)=\sum_{i=0}^n\kappa_{n-i}V_i(K)t^{n-i}\quad(t\geq0), \] where $B^n$ is the Euclidean unit ball and $\kappa_j$ its $j$-dimensional volume, with $\kappa_0=1$. Thus $V_n$ is volume, $V_{n-1}$ is half the boundary area in this convention, and $V_0(K)=1$.

The source question is whether \[ \forall n\geq2\quad\exists c(n)\in[0,\infty)\quad \forall K\subset{\mathbb{R}}^n\text{ convex body},\qquad G(K)\leq V_n(K)+V_{n-1}(K)+c(n)\sum_{i=0}^{n-2}V_i(K). \] The same finite constant must work for all bodies in a fixed dimension, including all translations; it may not depend on the body's size, shape, or position. The coefficients of the two highest intrinsic volumes are exactly one. No lattice-polytope, symmetry, or origin-containment assumption is present. For $n=1$, if the empty lower sum is interpreted as zero, the analogous displayed formula reads $G(K)\leq V_1(K)+V_0(K)$; the source does not discuss its dimension convention explicitly. Whether lower-dimensional convex sets are also included under "convex body" must be confirmed.

The supplied literature assessment checked 2026-08-21 says that the full Wills conjecture has high-dimensional counterexamples, but that this weaker inequality with a dimension-dependent coefficient on the lower intrinsic volumes remains open in the located literature. Failure of a bound with coefficient one on every intrinsic volume does not itself decide the weaker bound above.

Short English statement

Can lattice-point counts be bounded by volume plus the next intrinsic volume with coefficient one, allowing a dimension-dependent multiplier only on the remaining intrinsic volumes? The multiplier must work uniformly over all convex bodies in that dimension.

Sources

• Local UnsolvedMath registry, permanent ID 30000114, catalogue entry, A Weakened Wills Inequality for Convex Bodies.
• Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry (2005), Oberwolfach Reports, pp. 2071--2102, DOI https://doi.org/10.4171/owr/2004/39.
• Supplied reference labeled "Related Wills inequalities": https://epubs.siam.org/doi/pdf/10.1137/15M1031369?download=true.
• Supplied literature triage checked 2026-08-21 and supplied claim of statement comparison with the official report; no theorem numbers or full secondary citation are provided.