Abstract

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.

Author information

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Keywords

  • Variational analysis
  • amenable set
  • approximate convexity
  • normal embedding
  • prox-regularity

Mathematics Subject Classification

49J53, 90C31, 32C09, 51F30

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Suggested citation

A. S. Lewis, A. Nicolae, T. Tian. “Local Geometry of Feasible Regions via Smooth Paths.” Journal of Convex Analysis 32 (2025), No. 2, 511–544. https://doi.org/10.68381/jca32025

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.