On some relations between the perimeter, the area and the visual angle of a convex set
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J. Bruna
Abstract
We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton’s formula is the unique universal formula relating the visual angle, the length and the area. After that we give a characterization of convex sets of constant width by means of the behavior of their isotopic sets at infinity. Also for this class of convex sets we prove that the existence of an isotopic circle is enough to ensure that the considered set is a disc.
Funding statement: J. Bruna was partially supported by grants 2021SGR0087 (Generalitat de Catalunya) and PID2021-123405NB-I00 (Ministerio de Ciencia e Innovación).
Acknowledgements
The authors are grateful to A. Gasull for various conversations on the subject that have contributed to the proof of Theorem 4. We also thank E. Gallego for his useful comments.
Communicated by: M. Henk
References
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- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
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Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
