An isoperimetric inequality for logarithmic capacity
نویسندگان
چکیده
We prove a sharp lower bound of the form capE ≥ (1/2)diamE · Ψ(areaE/((π/4)diam 2E)) for the logarithmic capacity of a compact connected planar set E in terms of its area and diameter. Our lower bound includes as special cases G. Faber’s inequality capE ≥ diamE/4 and G. Pólya’s inequality capE ≥ (areaE/π)1/2. We give explicit formulations, functions of (1/2)diamE, for the extremal domains which we identify. 1 2
منابع مشابه
An isoperimetric inequality for logarithmic capacity of polygons
We verify an old conjecture of G. Pólya and G. Szegő saying that the regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area.
متن کاملLogarithmic Sobolev, Isoperimetry and Transport Inequalities on Graphs
In this paper, we study some functional inequalities (such as Poincaré inequality, logarithmic Sobolev inequality, generalized Cheeger isoperimetric inequality, transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of (random) path method. We provide estimates of the involved constants.
متن کاملSpectral Gap, Logarithmic Sobolev Constant, and Geometric Bounds
– We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We exami...
متن کاملA converse to Maz’ya’s inequality for capacities under curvature lower bound
We survey some classical inequalities due to Maz’ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz’ya’s lower bound for the q-capacity (q > 1) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show tha...
متن کاملLog-sobolev, Isoperimetry and Transport Inequalities on Graphs
In this paper, we study some functional inequalities (such as Poincaré inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method. We provide estimates of the involved c...
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 2002