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Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

Qing Liu, Atsushi Nakayasu
2019 Discrete and Continuous Dynamical Systems. Series A  
We study the convexity preserving property for a class of timedependent Hamilton-Jacobi equations in a complete geodesic space.  ...  Assuming that the Hamiltonian is nondecreasing, we show that in a Busemann space the unique metric viscosity solution preserves the geodesic convexity of the initial value at any time.  ...  Thanks to Proposition 2.7 it follows that u(·, t) is pointwise convex, as defined in Definition 2.8, for all time t ≥ 0.  ... 
doi:10.3934/dcds.2019007 fatcat:72avxkeonnhfxhu2gtwxbhy6oa

A note on graph drawings with star-shaped boundaries in the plane [article]

Yanwen Luo
2022 arXiv   pre-print
Moreover, we study the homotopy property of spaces of all straight-line embeddings. We give a simple argument to show that this space is contractible if the boundary is a non-convex quadrilateral.  ...  It is based on minimizing discrete Dirichlet energies, following the idea of Tutte's embedding theorem. We will call it a geodesic triangulation of the star-shaped polygon.  ...  Example in Contractibility of spaces of geodesic triangulations The field of discrete differential geometry features the discretization of the whole theory of classical geometry and topology of surfaces  ... 
arXiv:2204.10831v1 fatcat:cqnyp7vdbrg4tczhqgetkflnsu

The Bayesian update: variational formulations and gradient flows [article]

Nicolas Garcia Trillos, Daniel Sanz-Alonso
2018 arXiv   pre-print
We show that, in all cases, the rate of convergence of the flows to the posterior can be bounded by the geodesic convexity of the functional to be minimized.  ...  These diffusions may be discretized to build proposals for Markov chain Monte Carlo (MCMC) algorithms.  ...  Data" that took place at Carnegie Mellon University in March 2017.  ... 
arXiv:1705.07382v2 fatcat:nnc4yuzivnfkzhpkzculhgobey

Bézier curves in the space of images [article]

Alexander Effland, Martin Rumpf, Stefan Simon, Kirsten Stahn, Benedikt Wirth
2015 arXiv   pre-print
Geodesics are approximated using a variational discretization of the Riemannian path energy.  ...  This leads to a generalized de Casteljau method to compute suitable discrete B\'ezier curves in image space. Selected test cases demonstrate qualitative properties of the approach.  ...  Hence, these geodesics are not only the obvious generalization of straight lines in Euclidian space, but also allow a simple procedure to compute convex combinations with convex coefficients t and 1 −  ... 
arXiv:1503.02527v1 fatcat:aer6d3j5kvblnn35dafsskjd3a

On the isometry groups of certain CAT(0) spaces and trees

Daniel S. Farley
2000 Topology and its Applications  
More generally, we demonstrate that for certain proper CAT(0) spaces X, the group of isometries of X is not an inverse limit of Lie groups.  ...  We show that the automorphism group of a locally finite tree is discrete, or pro-finite, or not the inverse limit of an inverse system of discrete groups, and provide necessary and sufficient conditions  ...  I am grateful to the referee for pointing out an error in an earlier draft of this paper and suggesting possible corrections.  ... 
doi:10.1016/s0166-8641(99)00143-1 fatcat:shxcdsg6rzcp7ge7jaguu7qlly

Discrete-time gradient flows in Gromov hyperbolic spaces [article]

Shin-ichi Ohta
2022 arXiv   pre-print
We investigate fundamental properties of the proximal point algorithm for Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces.  ...  I would like to thank Hiroshi Hirai for his comments on convex functions on discrete spaces.  ...  In the sequel, however, we do not consider discrete spaces, mainly due to the difficulty of dealing with convex functions (see Subsection 3.4).  ... 
arXiv:2205.03156v1 fatcat:teoekkdxk5d2jhl43vblitvtxq

Long time behavior of quasi-convex and pseudo-convex gradient systems on Riemannian manifolds

P. Ahmadi, H. Khatibzadeh
2017 Filomat  
As an application to minimization, we consider a discrete version of the system to approximate a minimum point of a given pseudo-convex function ϕ.  ...  In this paper, we study the following gradient system on a complete Riemannian manifold M, where ϕ : M → R is a C 1 function with Argminϕ ∅.  ...  Consider the constrained minimization problem: Min x∈M ϕ(x). (3) In some cases ϕ is not quasi-convex on the whole space H, but it becomes quasi-convex (or even convex) on the constrained set M along geodesics  ... 
doi:10.2298/fil1714571a fatcat:eybp7wid6vgntmi6uq4l7sc5ry

2D and 3D visibility in discrete geometry: an application to discrete geodesic paths

D Coeurjolly, S Miguet, L Tougne
2004 Pattern Recognition Letters  
Based on these definitions, we define discrete geodesic paths in discrete domain with obstacles. This allows us to introduce a new geodesic metric in discrete geometry.  ...  We present efficient algorithms to compute the set of pixels in a non-convex domain that are visible from a source pixel.  ...  discrete straight segment from s to t whose pixels belong to D Before introducing the visibility problem in non-convex domain, we recall classical parameter space characterizations of DSL [19, 20, 32  ... 
doi:10.1016/j.patrec.2003.12.002 fatcat:ig3trqlicveljj24zwssuqawzm

Lecture Notes on Gradient Flows and Optimal Transport [article]

Sara Daneri, Giuseppe Savaré
2010 arXiv   pre-print
We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces  ...  These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.  ...  Generation results for geodesically convex functionals in spaces with a semiconcave squared distance.  ... 
arXiv:1009.3737v1 fatcat:bgiwzt537bbt3epk4z77ks27lu

Convex geodesic bicombings and hyperbolicity [article]

Dominic Descombes, Urs Lang
2014 arXiv   pre-print
) convex geodesic bicombing of the strongest type.  ...  A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them.  ...  Suppose that X is a complete metric space with a geodesic bicombing σ that is conical and 1/n-discretely convex for some integer n ≥ 2.  ... 
arXiv:1404.5051v1 fatcat:bmcljxyuwfeyfmeputwo64vfgy

Gradient flow structures for discrete porous medium equations [article]

Matthias Erbar, Jan Maas
2012 arXiv   pre-print
We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.  ...  This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in R^n discovered by Otto.  ...  In particular, H turns out to be convex along W-geodesics in one-dimensional discrete Fokker-Planck equations [13] as well as in heat equations on d-dimensional square lattices in arbitrary dimension  ... 
arXiv:1212.1129v1 fatcat:ydaua3rdibc3zmrufajgtzxxsq

A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature [article]

Djordje Baralic, Ioana-Claudia Lazar
2014 arXiv   pre-print
The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of the complex onto a ball around this convex subcomplex.  ...  The image α of c is called a geodesic segment with endpoints x and y. A geodesic metric space (X, d) is a metric space in which every pair of points can be joined by a geodesic segment.  ...  Discrete Morse theory The main tool we shall use in the proof is discrete Morse theory.  ... 
arXiv:1403.4547v1 fatcat:7b6quis4pfb7xae63dkuoi5nt4

CAT(0) groups with specified boundary

Kim Ruane
2006 Algebraic and Geometric Topology  
We specify exactly which groups can act geometrically on CAT(0) spaces whose visual boundary is homeomorphic to either a circle or a suspension of a Cantor set.  ...  Then G acts discretely, cocompactly, and isometrically on a convex subset of H 3 with nonempty totally geodesic boundary.  ...  Recall that a CAT(0) space X has local extendability of geodesics if every geodesic segment in X can be extended to a geodesic line in X.  ... 
doi:10.2140/agt.2006.6.633 fatcat:mnm3jljx4vgsfcril5dxpvy6ee

Discrete-time gradient flows and law of large numbers in Alexandrov spaces [article]

Shin-ichi Ohta, Miklós Pálfia
2015 arXiv   pre-print
We develop the theory of discrete-time gradient flows for convex functions on Alexandrov spaces with arbitrary upper or lower curvature bounds.  ...  We also prove a stochastic version, a generalized law of large numbers for convex function valued random variables, which not only extends Sturm's law of large numbers on nonpositively curved spaces to  ...  Acknowledgment The authors would like to thank the anonymous referee for his valuable comments, in particular improving the discussion in section 6.  ... 
arXiv:1402.1629v2 fatcat:ekpipnhq2bbqbhi4a5bk4fiely

Energy of Twisted Harmonic Maps of Riemann Surfaces [article]

William M. Goldman, Richard A. Wentworth
2006 arXiv   pre-print
More generally, if ρ is a discrete embedding onto a normal subgroup of a convex cocompact group Γ, then E_ρ defines a proper function on the quotient /Q where Q is the subgroup of the mapping class group  ...  The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface S is a function E_ρ on Teichmüller space which is a qualitative invariant of the holonomy  ...  In particular we thank Ian Agol, Francis Bonahon, Dick Canary, David Dumas, Cliff Earle, Joel Hass, Misha Kapovich, Bruce Kleiner, François Labourie, John Loftin, Yair Minsky, Rick Schoen, Bill Thurston  ... 
arXiv:math/0506212v2 fatcat:a5bngted3jfobozhmz4ktpwmqq
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