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Scott's Inequality

Scott's inequality bounds the number B of boundary lattice points of a convex lattice polygon in terms of its number I of interior lattice points. For I>=1, it gives B<=2I+7, with equality only for the exceptional triangle equivalent under an affine unimodular transformation to conv{(0,0),(3,0),(0,3)} (Scott 1976).

For a lattice polygon, the numerator of the Ehrhart polynomial series is h^*(t)=1+(B+I-3)t+It^2. Thus for I>=2, Scott's inequality implies normalized area at most 4(I+1). Treutlein (2010) extended this degree-two perspective to lattice polytopes. For a d-dimensional lattice polytope P, define h_P^*(t) by

sum_(m>=0)|mP intersection Z^d|t^m=(h_P^*(t))/((1-t)^(d+1)).

Tsuchiya (2026) proved that, if h_P^*(t)=1+at^k+bt^(2k) with a>=1, b>=2, and k>=1, then the normalized volume satisfies Vol(P)<=4k(b+1). Equality holds iff k=1 and a=3b+3. Here normalized volume is d! times the ordinary volume measured in the lattice. Tsuchiya (2026) reports AI assistance with the mathematical proof.

See also

Ehrhart Polynomial, Lattice Polygon

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References

Scott, P. R. "On Convex Lattice Polygons." Bull. Austral. Math. Soc. 15, 395-399, 1976. https://doi.org/10.1017/S0004972700022826.Treutlein, J. "Lattice Polytopes of Degree 2." J. Combin. Th., Ser. A 117, 354-360, 2010. https://doi.org/10.1016/j.jcta.2009.07.006.Tsuchiya, A. "Scott-Type Inequalities for Trinomial h^*-Polynomials." 26 Oct 2026. https://arxiv.org/abs/2610.35287.

Cite this as:

Weisstein, Eric W. "Scott's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ScottsInequality.html

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