2026-04-13T03:14:01Zhttps://recercat.cat/oai/request
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RECERCAT author Cufí Sobregrau, Julià, author Gallego Gómez, Eduardo author Reventós, Agustí 2018 Given a simple closed plane curve Γ of length L enclosing a compact convex set K of area F, Hurwitz found an upper bound for the isoperimetric deficit, namely L2-4πF≤π|Fe|, where Fe is the algebraic area enclosed by the evolute of Γ. In this note we improve this inequality finding strictly positive lower bounds for the deficit π|Fe|-Δ, where Δ=L2-4πF. These bounds involve either the visual angle of Γ or the pedal curve associated to K with respect to the Steiner point of K or the L2 distance between K and the Steiner disk of K. For compact convex sets of constant width Hurwitz's inequality can be improved to L2-4πF≤[Formula presented]π|Fe|. In this case we also get strictly positive lower bounds for the deficit [Formula presented]π|Fe|-Δ. For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of 3, 4 or 5 cusps or the Minkowski sum of this kind of sets. open access Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades. https://creativecommons.org/licenses/by-nc-nd/4.0/ Convex set Evolute Hypocycloid Isoperimetric inequality Pedal curve Visual angle A note on Hurwitz's inequality Article