Sharp inequalities involving the Cheeger constant of planar convex sets
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917600251871232 |
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| author | Ftouhi, Ilias Masiello, Alba Lia Paoli, Gloria |
| author_facet | Ftouhi, Ilias Masiello, Alba Lia Paoli, Gloria |
| contents | We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $ω$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(ω,h,d)$, $(ω,h,R)$, $(ω,h,P)$, $(ω,h,|\cdot|)$, $(R,h,d)$ and $(ω,h,r)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_13158 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp inequalities involving the Cheeger constant of planar convex sets Ftouhi, Ilias Masiello, Alba Lia Paoli, Gloria Analysis of PDEs 52A10, 52A40, 65K15 We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $ω$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(ω,h,d)$, $(ω,h,R)$, $(ω,h,P)$, $(ω,h,|\cdot|)$, $(R,h,d)$ and $(ω,h,r)$. |
| title | Sharp inequalities involving the Cheeger constant of planar convex sets |
| topic | Analysis of PDEs 52A10, 52A40, 65K15 |
| url | https://arxiv.org/abs/2206.13158 |