Abstract
Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets.
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Acknowledgements
The authors are indebted to the two referees, who not only carefully read an earlier draft of our paper, but also generously provided many insights, including examples and references, that significantly improved both the quality of this work and our understanding of the subject matter. We also thank Hongzhi Liao (UNSW Sydney) for correcting several inaccuracies in our proofs. We are grateful to the Australian Research Council for financial support provided by means of the Discovery Project “An optimisation-based framework for non-classical Chebyshev approximation”, DP180100602.
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Millán, R.D., Roshchina, V. The Intrinsic Core and Minimal Faces of Convex Sets in General Vector Spaces. Set-Valued Var. Anal 31, 14 (2023). https://doi.org/10.1007/s11228-023-00671-6
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DOI: https://doi.org/10.1007/s11228-023-00671-6