AReverse Isoperimetri Inequality, Stability and Extremal Theorems for Plane Curves with Bounded Curvature
نویسندگان
چکیده
Stability and Extremal Theorems for Plane Curves with Bounded Curvature Ralph Howard and Andrejs Treibergs In this note we dis uss some elementary theorems about the relation between area and length of losed embedded plane urves with bounded urvature. Our main result (see Theorem 4.1) solves the extremal problem of whi h domain has largest boundary length among embedded disks in the plane whose boundary urvatures are uniformly bounded and whose area is xed and suÆ iently small.
منابع مشابه
Relative volume comparison theorems in Finsler geometry and their applications
We establish some relative volume comparison theorems for extremal volume forms of Finsler manifolds under suitable curvature bounds. As their applications, we obtain some results on curvature and topology of Finsler manifolds. Our results remove the usual assumption on S-curvature that is needed in the literature.
متن کاملIsoperimetric Inequalities for Wave Fronts and a Generalization of Menzin’s Conjecture for Bicycle Monodromy on Surfaces of Constant Curvature
The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length L bounding area A on a surface of constant curvature c, L2 ≥ 4πA− cA2 with equality holding only if the curve is a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic...
متن کاملRegularity Properties of a Certain Class of Surfaces
Introduction. Convex curves in the plane and convex surfaces in space have many at first sight unexpected regularity properties. The stringency of the convexity condition makes it desirable to find a geometric property of convex curves or surfaces which has similar analytical implications but applies to a wider class of geometrical objects. The first and most studied generalization of convexity...
متن کاملSOME THEOREMS ON THE costtA INEQUALITY*;1) BY
In this paper we consider subharmonic functions »gl in the unit disk whose minimum modulus and maximum modulus satisfy a certain inequality. We show the existence of an extremal member of this class with largest maximum modulus. We then obtain an upper bound for the maximum modulus of this function in terms of the logarithmic measure of a certain set. We use this upper bound to prove theorems a...
متن کاملSharp Stability Theorems for the Anisotropic Sobolev and Log-sobolev Inequalities on Functions of Bounded Variation
Combining rearrangement techniques with Gromov’s proof (via optimal mass transportation) of the 1-Sobolev inequality, we prove a sharp quantitative version of the anisotropic Sobolev inequality on BV (R). As a corollary of this result, we also deduce a sharp stability estimate for the anisotropic 1-log-Sobolev inequality.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007