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Terracini Convexity

Saunderson, James and Chandrasekaran, Venkat (2020) Terracini Convexity. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20201016-144006753

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Abstract

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.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2010.00805arXivDiscussion Paper
ORCID:
AuthorORCID
Saunderson, James0000-0002-5456-0180
Additional Information:The authors would like to thank Rainer Sinn for helpful conversations. V.C. was supported in part by National Science Foundation grant CCF-1637598.
Funders:
Funding AgencyGrant Number
NSFCCF-1637598
Subject Keywords:face lattice, hyperbolic programming, moments and nonnegative polynomials, neighborly polytopes, semidefinite programming
Record Number:CaltechAUTHORS:20201016-144006753
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201016-144006753
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106117
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:16 Oct 2020 22:06
Last Modified:16 Oct 2020 22:06

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