Zaks–Perles–Wilkes volume conjecture for lattice polytopes

Let $d\geq 3$ and $k\geq 1$. A $k$-point polytope is a $d$-dimensional lattice polytope with $k$ interior lattice points. Define the Sylvester sequence by

$$s_1=2,\qquad s_i=s_1\cdots s_{i-1}+1,$$

and set

$$S_k^d:=\operatorname{conv}\{\boldsymbol{0},s_1e_1,\ldots,s_{d-1}e_{d-1},(k+1)(s_d-1)e_d\}.$$

Zaks–Perles–Wilkes volume conjecture. Every $k$-point $d$-dimensional lattice polytope $P$ satisfies

$$\operatorname{Vol}(P)\leq (k+1)(s_d-1)^2.$$

Except when $d=3$ and $k=1$, equality holds if and only if $P=S_k^d$.

The conjecture seeks a sharp bound for the volume of a lattice polytope in terms of its number of interior lattice points. The simplex $S_k^d$ was introduced by Zaks, Perles, and Wilkes as the proposed extremal example; the general inequality and its equality characterization remain open.