On the quantitative isoperimetric inequality in the plane with the barycentric distance

In this paper we study the following quantitative isoperimetric inequality in the plane: $\lambda_0^2(\Omega) \leq C \delta(\Omega)$ where $\delta$ is the isoperimetric deficit and $\lambda_0$ is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio $\delta(\Omega)/\lambda_0^2(\Omega)$ in the class of compact connected sets and in the class of convex sets

A semilinear problem with a W^{1,1}_0 solution

We study a degenerate elliptic equation, proving the existence of a W^{1,1}_0 distributional solution

Galactic Calibration of the Tip of the Red Giant Branch

Indications from Gaia data release 2 (DR2) are that the tip of the red giant branch (TRGB, a population II standard candle related to the helium flash in low mass stars) is close to -4 in absolute I magnitude in the Cousins photometric system. Our sample is high latitude southern stars from the thick disk and inner halo, and our result is consistent with longstanding findings from globular clusters, whose distances were calibrated with RR Lyrae stars. As the Gaia mission proceeds, there is every reason to think an accurate Galactic geometric calibration of TRGB will be a significant outcome.Comment: to appear in PAS