Abstract In spite of the geometric and analytic importance of hypocycloids, there are little systematic studies of geometric inequalities characterizing hypocycloids. Recently, Kwong--Lee obtained a countable family of geometric inequalities for convex curves by virtue of higher order Wirtinger-type inequalities. This family contains the isoperimetric inequality and Lin-Tsai's inequality. In this paper, generalizing the Kwong--Lee's family of inequalities to curves with admissible singularities (called ℓ-convex Legendre curves), we derive a countable family of inequalities whose equality conditions are characterized by hypocycloids. 2020 Mathematics Subject Classification. Primary 53A04; Secondary 52A40, 26D10.
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Honda et al. (2024) studied this question.
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