Journal of Convex Analysis 32 (2025), No. 2, 511--544
Copyright Heldermann Verlag 2025
Local Geometry of Feasible Regions via Smooth Paths
Adrian S. Lewis
ORIE, Cornell University, Ithaca, U.S.A.
adrian.lewis@cornell.edu
Adriana Nicolae
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
anicolae@math.ubbcluj.ro
Tonghua Tian
ORIE, Cornell University, Ithaca, U.S.A.
tt543@cornell.edu
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case of feasible regions consisting of pre-images of convex sets under maps that are once (but not necessarily twice) continuously differentiable, the geometry is less transparent. We define a new approximate convexity property, that holds both for such feasible regions and also for all prox-regular sets. This new property requires that nearby points can always be joined by smooth feasible paths that are almost straight. In particular, in the terminology of real algebraic geometry, such feasible regions are locally normally embedded in the Euclidean space.
Keywords: Variational analysis, amenable set, approximate convexity, normal embedding, prox-regularity.
MSC: 49J53, 90C31, 32C09, 51F30.
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