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Published June 2014 | public
Journal Article Open

From Steiner Formulas for Cones to Concentration of Intrinsic Volumes


The intrinsic volumes of a convex cone are geometric functionals that return basic structural information about the cone. Recent research has demonstrated that conic intrinsic volumes are valuable for understanding the behavior of random convex optimization problems. This paper develops a systematic technique for studying conic intrinsic volumes using methods from probability. At the heart of this approach is a general Steiner formula for cones. This result converts questions about the intrinsic volumes into questions about the projection of a Gaussian random vector onto the cone, which can then be resolved using tools from Gaussian analysis. The approach leads to new identities and bounds for the intrinsic volumes of a cone, including a near-optimal concentration inequality.

Additional Information

© 2014 Springer Science+Business Media New York. Received: 23 August 2013; Revised: 27 March 2014; Accepted: 1 April 2014. The authors thank Dennis Amelunxen and Martin Lotz for inspiring conversations and for their thoughtful comments on this material. This research was supported by ONR awards N00014-08-1-0883 and N00014-11-1002, AFOSR award FA9550-09-1-0643, and a Sloan Research Fellowship.

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August 20, 2023
August 20, 2023