Some results on curves in the plane with log-linear density

Abstract

In this paper, we show some relations between traveling fronts of curvature flow and constant [Formula: see text]-curvature curves. In the plane with density [Formula: see text] satisfying [Formula: see text], we prove a Fenchel’s type theorem for the class of simple, closed, convex curves. We also prove that among all curves joining two given points, a curve minimizes weighted arc-length if and only if its [Formula: see text]-curvature is 0. Another proof is also presented for the fact “the plane with density [Formula: see text] contains no isoperimetric region”.