For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the \(h^{*}\)-vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point).

对于具有至少一个内部晶格点的凸格多面体，其体积的下界是从针对（（h ^ {*} \））-向量的希比下界定理得出的。另一方面，已知极化变种的截面属具有上限，该上限是针对投射曲线属的Castelnuovo界的延伸。在本文中，我们证明了这两个界限的等价性。即，当且仅当与其相关的多表位具有最小体积时，极化复曲面变种才具有最大的截面属。这是对已知事实的概括，与戈伦斯坦复曲面法诺变种的反规范束相对应的多表位是反射型多表位（其典型示例是仅具有一个内部晶格点的最小体积的多表位）。