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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

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  • $\begingroup$ A wrong condition has been fixed. $\endgroup$
    – RyanChan
    Jan 25, 2021 at 0:02
  • $\begingroup$ Where did the problem arise? Why are you applying log-concave measure to the lattice sequence? $\endgroup$
    – Turbo
    Jan 25, 2021 at 2:49
  • $\begingroup$ Wouldn’t it be easier to say “let $g$ be a symmetric concave function”, and replace all instances of $\log f$ with $g$? The question never seems to use $f$ by itself. $\endgroup$
    – user44143
    Jan 25, 2021 at 3:54
  • $\begingroup$ @MattF. You are right. I have edited it according to your comment. $\endgroup$
    – RyanChan
    Jan 25, 2021 at 4:04
  • $\begingroup$ @1.. This problem arises in the areas of coding theory and information theory. $\endgroup$
    – RyanChan
    Jan 25, 2021 at 4:05

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