Concentration of lattice polytopes near the boundary volume

Let $B_r^d$ be the convex body used in the paper, and let $H_r^d$ be the set of lattice polytopes contained in $B_r^d$, with $H_r^d(V/d!\geqslant\mathrm{vol}(B_r^d)-c(r))$ denoting the subset whose volume is at least $\mathrm{vol}(B_r^d)-c(r)$. Then the boundary-volume concentration conjecture. There exists a function $c(r)=o(r^d)$ such that

$$\lim_{r\rightarrow\infty}\frac{\log |H_r^d\left(V/d!\geqslant \mathrm{vol}(B_r^d)-c(r)\right)|}{\log H_r^d}=1.$$

This predicts that, logarithmically, almost all lattice polytopes in $B_r^d$ have volume within $c(r)$ of the volume of $B_r^d$. The preceding results establish many such polytopes and indicate that the containment constraint has little effect on the logarithmic order, but the conjecture itself is unresolved in the supplied text.