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Hyperbolic polynomials have their origins in partial diierential equations. We show in this paper that they have applications in interior point methods for convex programming. Each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone. The function F (x) = ? log p(x) is a logarithmically homogeneous… (More)

We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for… (More)

A homogeneous polynomial p(x) is hyperbolic with respect to a given vector d if the real polynomial t 7 ! p(x + td) has all real roots for all vectors x. We show that any symmetric convex function of these roots is a convex function of x, generalizing a fundamental result of G arding. Consequently we are able to prove a number of deep results about… (More)

We characterize the barrier parameter of the optimal self{concordant barriers for homogeneous cones. In particular, we prove that for homogeneous convex cones this parameter is the same as the rank of the corresponding Siegel domain. We also provide lower bounds on the barrier parameter in terms of the Carath eodory number of the cone. The bounds are tight… (More)

A convex body K in R n has around it a unique circumscribed ellipsoid CE(K) with minimum volume, and within it a unique inscribed ellipsoid IE(K) with maximum volume. The modern theory of these ellipsoids is pioneered by Fritz John in his seminal 1948 paper. This paper has two, related goals. First, we investigate the symmetry properties of a convex body by… (More)