A Reverse Isoperimetric Inequality for Convex Plane Curves∗

A New Reverse Isoperimetric Inequality and Its Stability

In this paper, we deal with the reverse isoperimetric inequality for a closed and strictly convex curve in the Euclidean plane R2 involving the following geometric functionals associated to the given convex curve: length, areas of the region respectively included by the curve and the locus of curvature centers, and the integral of the radius of curvature. In fact, a stronger and sharp version o...

Steiner Polynomials, Wulff Flows, and Some New Isoperimetric Inequalities for Convex Plane Curves

Stability results for some geometric inequalities and their functional versions ∗

The Blaschke Santaló inequality and the Lp affine isoperimetric inequalities are major inequalities in convex geometry and they have a wide range of applications. Functional versions of the Blaschke Santaló inequality have been established over the years through many contributions. More recently and ongoing, such functional versions have been established for the Lp affine isoperimetric inequali...

Isoperimetric inequalities in the Heisenberg group and in the plane

We formulate the isoperimetric problem for the class of C2 smooth cylindrically symmetric surfaces in the Heisenberg group in terms of Legendrian foliations. The known results for the sub-Riemannian isoperimetric problem yield a new isoperimetric inequality in the plane: For any strictly convex, C2 loop γ ∈ R2, bounding a planar region ω, we have

Minkowski isoperimetric-hodograph curves

General offset curves are treated in the context of Minkowski geometry, the geometry of the two-dimensional plane, stemming from the consideration of a strictly convex, centrally symmetric given curve as its unit circle. Minkowski geometry permits us to move beyond classical confines and provides us with a framework in which to generalize the notion of Pythagorean-hodograph curves in the case o...