Mathematical Programming

, Volume 149, Issue 1–2, pp 105–130 | Cite as

Intrinsic volumes of symmetric cones and applications in convex programming

  • Dennis AmelunxenEmail author
  • Peter Bürgisser
Full Length Paper Series A


We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions in terms of integrals related to Mehta’s integral. In particular, we obtain a closed formula for the probability that the solution of a random (standard Gaussian) semidefinite program has a certain rank.


Random convex programs Semidefinite programming  Intrinsic volumes Symmetric cones Mehta’s integral 

Mathematics Subject Classification (2000)

15B48 52A55 53C65 60D05 90C22 



We thank Michael B. McCoy for pointing out that almost sure nonvanishing is the only assumption on \(b\) that is needed in a standard Gaussian (CP). We are grateful to the anonymous referees for comments that led to a more structured presentation. This work has been supported by the grants AM 386/1-1 and BU 1371/2-2 of the German Research Foundation (DFG).


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.School of MathematicsThe University of ManchesterManchesterUK
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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