How does the contraction property fail for convex functions on normed spaces?

Shin-ichi Ohta

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arXiv · 2311.15152

How does the contraction property fail for convex functions on normed spaces?

Shin-ichi Ohta

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Abstract

On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT$(0)$-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces.

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Shin-ichi Ohta. 2023-11-26. How does the contraction property fail for convex functions on normed spaces?. https://arxiv.org/abs/2311.15152

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