The Engulfing Property from a Convex Analysis Viewpoint
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The Engulfing Property from a Convex Analysis Viewpoint Andrea Calogero1 · Rita Pini1 Received: 17 October 2019 / Accepted: 11 September 2020 / Published online: 28 September 2020 © The Author(s) 2020
Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property. Keywords Engulfing property · Soft engulfing property · Convex function Mathematics Subject Classification 26B25 · 26A12
1 Introduction The convex functions satisfying the so-called engulfing property have been studied in connection with the solution to the Monge–Ampère equation. Several conditions on such functions have been proposed in order to preserve the harmony between measure theory (and, in particular, the Monge–Ampère measure related to a convex function) and the shape of the sections, with their induced geometry; in this framework, we would like to mention the celebrated C 1,β -estimate due to Caffarelli [1,2], and the exhaustive book by Gutiérrez [3]. This study involved many authors with different points of view; very interesting are the papers by Gutiérrez and Huang [4], and by Forzani and Maldonado [5,6]. Let us devote a few lines to the mentioned C 1,β -estimate: One of the proposed conditions on a convex function, with bounded sections, is the so-called (DC)-doubling property of the related Monge–Ampère measure (for details, see, for example, [6]); this condition plays a fundamental role in the whole theory of the Monge–Ampère equation, since it is equivalent to the engulfing property of the
Communicated by Julian P. Revalski.
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Rita Pini [email protected] Andrea Calogero [email protected]
1
University of Milano-Bicocca, Milano, Italy
123
Journal of Optimization Theory and Applications (2020) 187:408–420
409 1,β
related function, and it implies that this function is strictly convex and in Cloc . Due to this equivalence, the study of the engulfing property of a convex function is often moved to the study of the regularity of the related Monge–Ampère measure. The purpose of this note, differently from the literature, is to bring into focus that strict convexity and differentiability are properties intrinsic to the engulfing. This is done by means of purely convex analytic elementary techniques, and without taking into account the properties of the related Monge–Ampère measure. The paper is organized as follows: In Sect. 2, we recall the notion of engulfing for a convex function and introduce the notion of soft engulfing. We provide a fine monotonicity result for the subdifferential map of a convex function that enjoys the soft engulfing property. Furthermore, we prove that this property of the function entails continuous differentiability, as well as strict convexity. In Sect. 3, we prove the equivalence between the class of functions satisfying the engulfing property and the class of fu
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