Random fluctuations of convex domains and lattice points

In this paper, we examine a random version of the lattice point problem. Let $\mathcal H$ denote the class of all homogeneous functions in $C^2(\mathbb R^n)$ of degree one, positive away from the origin. Let $\Phi$ be a random element of $\mathcal H$, defined on probability space $(\Omega,\mathcal F,P)$, and define

for $\omega\in\Omega$. We prove that, if $E(|F_\Phi(\xi)|)\le C[\xi]^{\frac{n+1}{2}}$, where $[\xi]=1+|\xi|$, then

where $V=E(|\{x\colon\Phi(\cdot,x)\le 1\}|)$, the expected volume. That is, on average, $N_\Phi(t)=Vt^n+O(t^{n-2+\frac{2}{n+1}})$. We give explicit examples in which the Gaussian curvature of $\{x\colon \Phi(\omega,x)\le 1\}$ is small with high probability, and the estimate $N_\Phi(t)=Vt^n+O(t^{n-2+\frac{2}{n+1}})$ holds nevertheless.