DOI: 10.68381/jca22043 ISSN: 0944-6532
A Sharp Quantitative Estimate for the Perimeters of Convex Sets in the Plane
Menita Carozza, Flavia Giannetti, Francesco Leonetti, Antonia Passarelli di Napoli
Let
E \subset B \subset \mathbb R^2
E
⊂
B
⊂
R
2
be bounded, convex sets. The monotonicity of the perimeters holds, i.e.
\mathcal{H}^{1} (\partial E) \leqslant \mathcal{H}^{1}(\partial B)
H
1
(
∂
E
)
⩽
H
1
(
∂
B
)
. Here we give a quantitative estimate of the difference of the perimeters: it shows how much the perimeter of
B
B
increases when
B
B
becomes larger and larger with respect to
E
E
. We give an example showing that our estimate is sharp.