Extremal problems for convex lattice polytopes

In this survey talk I will present several extremal problems, and some solutions, concerning convex lattice polytopes. A polytope is called a lattice polytope if all of its vertices belong to the integer lattice $\mathbb Z^d$. Let $\mathcal P(n,d)$ denote the family of all convex lattice polytopes, of positive volume, in $\mathbb R^d$ with $n$ vertices. The following extremal problems will be considered.

• 1. minimal volume for $P\in\mathcal P(n,d)$,
• 2. minimal surface area for $P\in\mathcal P(n,d)$,
• 3. minimal lattice width for $P\in\mathcal P(n,d)$,
• 4. maximal $n$ such that a (large) convex set $K\subset\mathbb R^d$ contains and element of $\mathcal P(n,d)$, in other words, the maximal number of lattice points in $K$ that are in convex position.

These problems are related to a question of V. I. Arnold from 1980 asking for the number of (equivalence classes of) lattice polytopes of volume (at most) $V$ in $d$-dimensional space. Here two convex lattice polytopes are equivalent if one can be carried to the other by a lattice preserving affine transformation.