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Published October 16, 2020 | In Press + Submitted
Journal Article Open

Terracini convexity


We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive-semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems.

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© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received 19 November 2020; Accepted 09 January 2022; Published 05 March 2022. The authors would like to thank Rainer Sinn for helpful conversations, and the anonymous referees for their useful comments and suggestions. V. C. was supported in part by National Science Foundation grant CCF-1637598, in part by National Science Foundation grant DMS-2113724, and in part by AFOSR grant FA9550-20-1-0320. J. S. was supported in part by an Australian Research Council Discovery Early Career Researcher Award (project number DE210101056) funded by the Australian Government. Open Access funding enabled and organized by CAUL and its Member Institutions.

Attached Files

Submitted - 2010.00805.pdf

In Press - Saunderson-Chandrasekaran2022_Article_TerraciniConvexity.pdf


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August 22, 2023
October 20, 2023