Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

Abstract

Abstract. An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the L 1 case has drawn much less attention.In this note we prove two quantitative isoperimetric inequalities in the L 1 Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch. Statement of the resultsWe consider the plane R 2 endowed with the L 1 metricWe denote the boundary of a set by ∂. The notation | · | shall be used to denote the size of an object, whatever its nature. If A is a measurable plane set, then |A| is its Lebesgue measure, also called its area; if v is a vector, |v| is its L 1 norm; if γ is a curve, |γ| is its L 1 length, namelywhere the supremum is over all increasing sequences (t 1 , t 2 , . . . , t n ) taking values in the definition interval of γ. A curve is said to be (L 1 -)rectifiable if its length is finite. For example, a curve defined on a segment whose coordinate functions are both monotonic is rectifiable, and its length is the distance between its endpoints. See e.g. [AT04] for more information on the length of curves in a metric space.By a domain of the plane, we mean the closure of the bounded component of a Jordan curve. In particular, domains are compact and connected. All rectangles and squares considered are assumed to have their sides parallel to the coordinate