On Perimeters and Volumes of Fattened Sets
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 t...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2019-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2019/8283496 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |