Polar isoperimetry. I: The case of the plane (Q2190632)

The authors first show that for a smooth real-valued function \(f\) on \([0,1]\), the inequality \[\int_0^1{\sqrt{f(x)^2+\frac{1}{\pi^2}f'(x)^2}dx}\geq \left(\int_0^1{f(x)^2dx}\right)^{1/2}\] is a consequence of the isoperimetric inequality in the upper half-plane \(\mathbb R_{+}^2\) \[\mu^{+}(A)\geq\sqrt{2\pi}(\mu(A))^{1/2}\] where \(\mu\) is the Lebesgue measure and \(\mu^{+}\) is the corresponding perimeter of a Borel measurable set \(A\subset\mathbb R_{+}^2\). Then they establish relationships with Poincaré-type inequalities and derive logarithmic Sobolev inequalities. In the last part of the paper, informational quantities and distances are discussed. For the entire collection see [Zbl 1431.60003].