Isoperimetric inequality

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  • We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular n-gon minimizes the logarithmic capacity among all n-gons with a fixed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is defined by (1.1) − log cap E = lim (g(z, ∞) − log |z|), z→∞ where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §11.1]. By an n-gon with...

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  • A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall. 1. Introduction The classical isoperimetric inequality states that if E is a Borel set in Rn , n ≥ 2, with finite Lebesgue measure |E|, then the ball with the same volume has a lower perimeter, or, equivalently, that (1.1) 1/n nωn |E|(n−1)/n ≤ P (E) . Here P (E) denotes the distributional perimeter of E (which coincides with the classical (n − 1)-dimensional measure of ∂E when E has a smooth boundary) and ωn is the measure of the...

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  • Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On Isoperimetric Inequalities in Minkowski Spaces

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