Let $f$ is a convex symmetric function on the interval $[-a,a]$, i.e., $f(-x)=f(x)$ for $\forall \, x\in [-a,a]$. Then we consider a $n$-dimensional convex body in Euclidean space \begin{equation} \mathscr{R}_n=\left\{ \mathbf{x}=\left(x_1,\cdots,x_n \right)\in \mathbb{R}^n： \sum_{i=1}^n f(x_i) \le n\mathsf{A} \right\}, \end{equation} where the constant $\mathsf{A}>f(0)$. Clearly, $\mathscr{R}_n$ is symmetric about the origin. I am interested the following problem relating to the lower bound of numbers of the lattice points inside $\mathscr{R}_n$:
Does there exit a lattice sequence $\left\{ \Lambda_n \right\}$ (the minimum distance of $ \Lambda_n$ is $\sqrt{n}$) such that the number sequence $\{N(n)\}$ satisfies \begin{equation} N(n)=\frac{\log \left( |\Lambda_n \cap \mathscr{R}_n| \right)}{n}\ge c, ~\text{for sufficiently large $n$.} \end{equation} I also want to know if there exits a best lowerbound $c$ for this asymptotic problem.
This problem is motivated by the answer of Geometry interpretation of any continuous random variable
