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We introduce a function w k (θ ) for closed convex plane curves, and then prove a geometric inequality involving w k (θ ) and the area A enclosed by the curve. As a by-product, we give a new proof of the classical isoperimetric inequality. Finally, we give some properties of convex curves with w k (θ ) being constant and pose an open problem motivated by the elegant Blaschke-Lebesgue theorem.

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An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of nonclassical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let G ba a connected semi-reductive algebraic group over an algebraically closed field F and g = Lie(G). It turns out that G has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for g, which says that the G-invariant ring F[g] G is a polynomial ring if g satisfies a certain "posivity" condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.