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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 L 2 − 4 π F ≤ π | F e | , where F e 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 | F e | − Δ , where Δ = L 2 − 4 π F . 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 L 2 distance between K and the Steiner disk of K . For compact convex sets of constant width Hurwitz's inequality can be improved to L 2 − 4 π F ≤ 4 9 π | F e | . In this case we also get strictly positive lower bounds for the deficit 4 9 π | F e | − 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. [ABSTRACT FROM AUTHOR]

Abstract: Using results from integral geometry, we find inequalities involving mean curvature integrals of convex hypersurfaces in hyperbolic space. Such inequalities generalize the Minkowski formulas for euclidean convex sets. [Copyright &y& Elsevier]