对数凹密度下$\mathbb{R}^1$上等周$n$ -泡问题的解
Solution to the isoperimetric $n$-bubble problem on $\mathbb{R}^1$ with log-concave density
摘要：
我们研究了$\mathbb{R}^1$上的等周问题，并给出了影响面积和周长测量的密度函数$f$。我们研究的密度函数是对称的，径向增加，并满足两个额外的条件：他们有一个点的零密度（在原点），他们满足“对数”的要求$\left[ \log f \right]'' \leq 0$。在这些条件下，我们发现等周$n$泡满足一个规则的结构，并可以识别为任意$n$。这概括了最近的工作做的密度函数$|x|^p$，并站在相反的对数凸密度函数，没有这样的规则结构。
英文摘要：
We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed density function $f$ that affects how area and perimeter are measured. We examine density functions that are symmetric, radially increasing, and satisfy two additional conditions: they have a point of zero density (at the origin), and they satisfy a "log-concavity" requirement $\left[ \log f \right]'' \leq 0$. Under these conditions, we find that isoperimetric $n$-bubbles satisfy a regular structure and can be identified for arbitrary $n$. This generalizes recent work done on the density function $|x|^p$, and stands in contrast to log-convex density functions which have no such regular structure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
