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For each dimension $d$, $d$-dimensional integral simplices with exactly oneinterior integral point have bounded volume. This was first shown by Hensley.Explicit volume bounds were determined by Hensley, Lagarias and Ziegler,Pikhurko, and Averkov. In this paper we determine the exact upper volume boundfor such simplices and characterize the volume-maximizing simplices. We alsodetermine the sharp upper bound on the coefficient of asymmetry of an integralpolytope with a single interior integral point. This result confirms aconjecture of Hensley from 1983. Moreover, for an integral simplex withprecisely one interior integral point, we give bounds on the volumes of itsfaces, the barycentric coordinates of the interior integral point and itsnumber of integral points. Furthermore, we prove a bound on the latticediameter of integral polytopes with a fixed number of interior integral points.The presented results have applications in toric geometry and in integeroptimization.

Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results