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38
Regularity of Horizons and The Area Theorem
"... We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperboli ..."
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Cited by 16 (12 self)
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We prove that the area of sections of future event horizons in spacetimes satisfying the null energy condition is nondecreasing towards the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic spacetime and there exists a conformal completion with a "regular" I + ; 3) the horizon is a black hole event horizon in a spacetime which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends ...
Structure of the unitary valuation algebra
 J. Differential Geom
"... Abstract. S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere S n−1 then the vector space Val G of continuous, translationinvariant, Ginvariant convex valuations on R n has the structure of a finite dimensional graded algebra over R satisfying Poincar ..."
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Cited by 9 (0 self)
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Abstract. S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere S n−1 then the vector space Val G of continuous, translationinvariant, Ginvariant convex valuations on R n has the structure of a finite dimensional graded algebra over R satisfying Poincaré duality. We show that the kinematic formulas for G are determined by the product pairing. Using this result we then show that the algebra Val U(n) is isomorphic to R[s, t]/(fn+1, fn+2), where s, t have degrees 2 and 1 respectively, and the polynomial fi is the degree i term of the power series log(1 + s + t). 1.
The lightcone theorem
, 2009
"... We prove that the area of crosssections of lightcones, in spacetimes satisfying suitable energy conditions, is smaller than or equal to that of the corresponding crosssections in Minkowski, or de Sitter, or antide Sitter spacetime. The equality holds if and only if the metric coincides with th ..."
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Cited by 9 (1 self)
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We prove that the area of crosssections of lightcones, in spacetimes satisfying suitable energy conditions, is smaller than or equal to that of the corresponding crosssections in Minkowski, or de Sitter, or antide Sitter spacetime. The equality holds if and only if the metric coincides with the corresponding model in the domain of dependence of the lightcone.
THE PROBABILITY THAT A SLIGHTLY PERTURBED NUMERICAL ANALYSIS PROBLEM IS DIFFICULT
, 2008
"... We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving ..."
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Cited by 8 (8 self)
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We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of illposed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of εtubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius σ. Besides ε and σ, this bound depends only on the dimension of the sphere and on the degree of the defining equations.
HERMITIAN INTEGRAL GEOMETRY
, 801
"... Abstract. We give in explicit form the principal kinematic formula for the action of the affine unitary group on C n, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, ..."
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Cited by 6 (2 self)
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Abstract. We give in explicit form the principal kinematic formula for the action of the affine unitary group on C n, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitaryinvariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements. 1.
Estimates on the distribution of the condition number of singular matrices
 Found. Comput. Math
"... We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional ..."
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Cited by 6 (4 self)
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We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equi–dimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.
General formulas for the smoothed analysis of condition numbers
, 2006
"... We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis e ..."
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Cited by 5 (3 self)
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We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis estimates for some conic condition numbers.
Robust smoothed analysis of a condition number of linear programming
, 2009
"... We perform a smoothed analysis of the GCCcondition number C(A) of the linear programming feasibility problem ∃x ∈ R m+1 Ax < 0. Suppose that Ā is any matrix with rows ai of euclidean norm 1 and, independently for all i, let ai be a random perturbation of ai following the uniform distribution in ..."
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Cited by 4 (4 self)
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We perform a smoothed analysis of the GCCcondition number C(A) of the linear programming feasibility problem ∃x ∈ R m+1 Ax < 0. Suppose that Ā is any matrix with rows ai of euclidean norm 1 and, independently for all i, let ai be a random perturbation of ai following the uniform distribution in the spherical disk in S m of angular radius arcsinσ and centered at ai. We prove that E(ln C(A)) = O(mn/σ). A similar result was shown for Renegar’s condition number and Gaussian perturbations by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius arcsinσ, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of Bürgisser, Cucker, and Lotz (Math. Comp. 77, No. 263, 2008) with techniques of Dunagan et al.
VOLUME MINIMIZATION AND ESTIMATES FOR CERTAIN ISOTROPIC SUBMANIFOLDS IN COMPLEX PROJECTIVE SPACES
, 2004
"... Abstract. In this note we show the following result using the integralgeometric formula of R. Howard: Consider the totally geodesic RP 2m in CP n. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CP n (but not among all submanifolds in this Z/2 homology cl ..."
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Cited by 4 (0 self)
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Abstract. In this note we show the following result using the integralgeometric formula of R. Howard: Consider the totally geodesic RP 2m in CP n. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CP n (but not among all submanifolds in this Z/2 homology class). Also the totally geodesic RP 2m−1 minimizes volume in its Hamiltonian deformation class in CP n. As a corollary we’ll give estimates for volumes of Lagrangian submanifolds in complete intersections in CP n. 1.
Probabilistic analysis of the Grassmann condition number
, 2013
"... We analyze the probability that a random mdimensional linear subspace of R n both intersects a regular closed convex cone C ⊆ R n and lies within distance α of an mdimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of ..."
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Cited by 4 (4 self)
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We analyze the probability that a random mdimensional linear subspace of R n both intersects a regular closed convex cone C ⊆ R n and lies within distance α of an mdimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number C (A) for the homogeneous convex feasibility problem ∃x ∈ C \ 0: Ax = 0. The Grassmann condition number is a geometric version of Renegar’s condition number, that we have introduced recently in [SIOPT 22(3):1029–1041, 2012]. We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of A ∈ R m×n are chosen i.i.d. standard normal, then for any regular cone C, we have E[ln C (A)] < 1.5 ln(n) + 1.5. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.