Local characterization of polyhedral spaces

Lebedeva, Nina; Petrunin, Anton

doi:10.1007/s10711-015-0072-x

Cited by 8 publications

(9 citation statements)

References 6 publications

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“…(If P is a spherical polyhedral space, then a small neighborhood of p is isometric to a neighborhood of the north pole in Susp Σ p .) In fact, if this property holds at any point of a compact length space P, then P is a polyhedral space, see [39] by Nina Lebedeva and the third author.…”

Section: Polyhedral Spacesmentioning

confidence: 99%

An Invitation to Alexandrov Geometry

Alexander

Kapovitch

Petrunin

2019

SpringerBriefs in Mathematics

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Section: Polyhedral Spacesmentioning

confidence: 99%

An Invitation to Alexandrov Geometry

Alexander

Kapovitch

Petrunin

2019

SpringerBriefs in Mathematics

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“…(3) For i = n−2, i.e. for the integral of the scalar curvature, the conjecture was claimed to be proven by A. Petrunin and N. Lebedeva in the work in progress [13].…”

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confidence: 97%

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Alesker

2018

Arnold Math J.

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For any closed smooth Riemannian manifold H. Weyl [21] has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called weakly smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, particularly the behavior under the Gromov-Hausdorff limits. We suggest conjectural compactifications of the space of smooth closed connected Riemannian manifolds with given upper bounds on dimension and diameter and a lower bound on sectional curvature to which the intrinsic volumes extend by continuity. We discuss also known cases of some of these conjectures. The work is a joint project with A. Petrunin.

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“…If is locally conical, i.e. is locally conical at every x ∈ , cannot have infinitely many vertices as a consequence of a general result in metric-space geometry [64].…”

Section: Definition 36 (Locally Dilation Invariant) When Is the Bound...mentioning

confidence: 99%

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

Chandler-Wilde

Spence

2021

Numer. Math.

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It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm $$<1/2$$ < 1 / 2 as an operator on the natural trace space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) whenever $$\Gamma $$ Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) for any sequence of finite-dimensional subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . Long-standing open questions are whether the essential norm is also $$<1/2$$ < 1 / 2 for D as an operator on $$L^2(\Gamma )$$ L 2 ( Γ ) for all Lipschitz $$\Gamma $$ Γ in 2-d; or whether, for all Lipschitz $$\Gamma $$ Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $$\pm \frac{1}{2}I+D$$ ± 1 2 I + D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$L^2(\Gamma )$$ L 2 ( Γ ) . We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is $$\ge 1/2$$ ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators $$\pm \frac{1}{2}I +D$$ ± 1 2 I + D are not coercive plus compact. We also give, for every $$C>0$$ C > 0 , examples of Lipschitz polyhedra for which the essential norm is $$\ge C$$ ≥ C and for which $$\lambda I+D$$ λ I + D is not a compact perturbation of a coercive operator for any real or complex $$\lambda $$ λ with $$|\lambda |\le C$$ | λ | ≤ C . We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the $$L^2(\Gamma )$$ L 2 ( Γ ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on $$C(\Gamma )$$ C ( Γ ) , equivalent to the standard supremum norm, for which the essential norm of D on $$C(\Gamma )$$ C ( Γ ) is $$<1/2$$ < 1 / 2 .

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Local characterization of polyhedral spaces

Abstract: We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.

Cited by 8 publications

References 6 publications

An Invitation to Alexandrov Geometry

An Invitation to Alexandrov Geometry

Some Conjectures on Intrinsic Volumes of Riemannian Manifolds and Alexandrov Spaces

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

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