On Perimeters and Volumes of Fattened Sets

Abstract EN

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C.

We provide two proofs: one elementary and one based on Geometric Measure Theory.

Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a).

We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(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.