Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

On Perimeters and Volumes of Fattened Sets

created by mennucci on 26 Apr 2019


Published Paper

Inserted: 26 apr 2019
Last Updated: 26 apr 2019

Journal: International Journal of Mathematics and Mathematical Sciences
Volume: 2019
Year: 2019
Doi: 10.1155/2019/8283496

(open access journal)


In this paper we analyze the shape of fattened sets; given a compact set $C\subset {\mathbb R}^N$ let $C_r$ be its $r-$fattened set, we prove a general bound $r\,P(C_r) \le N\, {\mathcal L}(\{C_r\setminus C\})$ between the perimeter of $C_r$ and the Lebesgue measure of $C_r\setminus C$. We provide two proofs, one elementary, and one based on Geometric Measure Theory. Note that, by the Flemin--Rishel coarea formula, $P(C_r)$ is integrable for $r\in(0,a)$. We further show that for any integrable continuous decreasing function $\psi:(0,1/2)\to(0,\infty)$ there exists a compact set $C\subset {\mathbb R}^N$ such that $P(C_r)\ge \psi(r)$. These results solve a conjecture left open in (Mennucci and Duci, 2015), and provide new insight in applications where the fattened set plays an important role.