E.B. Vinberg: The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12 (1963) 340-403.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Boundary Behaviour of the Riemannian Structure of a Self Concordant Barrier Function J.J. Duistermaat Department of Mathematics, Utrecht University, Postbus 80.010, 3508 TA Utrecht, The Netherlands. e mail: duis math.uu. nl August 19, 1999 1 Introduction Following Vinberg [18], we will define a convex domain in R as an open convex which does not contain a full straight line. A self concordant barrier function for a convex domain Q is defined as a strongly convex smooth function f on Q, which tends to 1 at the boundary and satisfies the estimates (iii) and (iv) of ....

....below, for the derivatives of f up to the third order. The strong convexity of f means that the Hessian g ij (x) i j f(x) is positive definite and therefore defines a Riemannian structure on Q. Such Riemannian structures on convex domains have been studied already by Koszul [13] and Vinberg [18], who refer further back to the theory of bounded domains in C , with its Bergmann metric. In this paper we investigate the asymptotic behaviour of this Riemannian structure, and of its geodesics and its curvature, near points of the boundary where the boundary is smooth and strongly convex, ....

E.B. Vinberg: The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12 (1963) 340-403.

....structures in Proposition 4.1. In the case n = 2 there is also an interpretation of the curvature of a Hessian Riemannian structure in terms of umbilic points of surfaces, see Section 5. The study of Hessian Riemannian structures on convex domains goes back at least to Koszul [6] and Vinberg [11], who were inspired by the theory of bounded domains in C with its Bergmann metric. Closely related to our subject is Shima s theory of Hessian manifolds, cf. 10] Ruuska [8] characterized Hessian Riemannian structures as those which admit an abelian Lie algebra of gradient vector fields, ....

E.B. Vinberg: The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12 (1963) 340-403. 15

....condition q 1 Gamma q 2 q 3 Gamma q 4 = 0 yields q 1 = q 2 and q 3 = q 4 . From SO(2) invariance we conclude q 1 q 2 Gamma q 3 Gamma q 4 = 0 which implies q 1 = q 2 = q 3 = q 4 . Since normal form transformations with no resonant terms preserve symmetries and reversibility (see [Brj71, Arn83]) we arrive at w j = w j [ j Phi j ( j; w 1 w 2 w 3 w 4 ) for j = 1; 4: 3.6) Using (w 3 ; w 4 ) w 1 ; w 2 ) and reversibility we find ( Phi 2 ; Phi 3 ; Phi 4 ) Gamma Phi 1 ; Phi 1 ; Gamma Phi 1 ) and thus (3.5) is established with Phi = Phi 1 . For C k vector fields ....

A.D. Brjuno: The analytical form of differential equations. Trans. Moscow Math. Society 25 (1971) 131-288.

....the Boundary Behaviour of the Riemannian Structure of a Self Concordant Barrier Function J.J. Duistermaat Department of Mathematics, Utrecht University, Postbus 80.010, 3508 TA Utrecht, The Netherlands. e mail: duis math.uu. nl August 19, 1999 1 Introduction Following Vinberg [18], we will define a convex domain in R n as an open convex subset of R n which does not contain a full straight line. A self concordant barrier function for a convex domain Q is defined as a strongly convex smooth function f on Q, which tends to 1 at the boundary and satisfies the estimates ....

....for the derivatives of f up to the third order. The strong convexity of f means that the Hessian g ij (x) i j f(x) is positive definite and therefore defines a Riemannian structure on Q. Such Riemannian structures on convex domains have been studied already by Koszul [13] and Vinberg [18], who refer further back to the theory of bounded domains in C n , with its Bergmann metric. In this paper we investigate the asymptotic behaviour of this Riemannian structure, and of its geodesics and its curvature, near points of the boundary where the boundary is smooth and strongly convex, ....

E.B. Vinberg: The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12 (1963) 340-403.

....structures in Proposition 4.1. In the case n = 2 there is also an interpretation of the curvature of a Hessian Riemannian structure in terms of umbilic points of surfaces, see Section 5. The study of Hessian Riemannian structures on convex domains goes back at least to Koszul [6] and Vinberg [11], who were inspired by the theory of bounded domains in C n with its Bergmann metric. Closely related to our subject is Shima s theory of Hessian manifolds, cf. 10] Ruuska [8] characterized Hessian Riemannian structures as those which admit an abelian Lie algebra of gradient vector fields, ....

E.B. Vinberg: The theory of convex homogeneous cones. Trans. Moscow Math. Soc. 12 (1963) 340-403.

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