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Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

2019
*
Discrete and Continuous Dynamical Systems. Series A
*

We study the

doi:10.3934/dcds.2019007
fatcat:72avxkeonnhfxhu2gtwxbhy6oa
*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. ...##
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A note on graph drawings with star-shaped boundaries in the plane
[article]

2022
*
arXiv
*
pre-print

Moreover, we study the homotopy property of

arXiv:2204.10831v1
fatcat:cqnyp7vdbrg4tczhqgetkflnsu
*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 ...##
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The Bayesian update: variational formulations and gradient flows
[article]

2018
*
arXiv
*
pre-print

We show that,

arXiv:1705.07382v2
fatcat:nnc4yuzivnfkzhpkzculhgobey
*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. ...##
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Bézier curves in the space of images
[article]

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 − ...

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

2000
*
Topology and its Applications
*

More generally, we demonstrate that for certain proper CAT(0)

doi:10.1016/s0166-8641(99)00143-1
fatcat:shxcdsg6rzcp7ge7jaguu7qlly
*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. ...##
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Discrete-time gradient flows in Gromov hyperbolic spaces
[article]

2022
*
arXiv
*
pre-print

We investigate fundamental properties of the proximal point algorithm for Lipschitz

arXiv:2205.03156v1
fatcat:teoekkdxk5d2jhl43vblitvtxq
*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). ...##
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Long time behavior of quasi-convex and pseudo-convex gradient systems on Riemannian manifolds

2017
*
Filomat
*

As an application to minimization, we consider a

doi:10.2298/fil1714571a
fatcat:eybp7wid6vgntmi6uq4l7sc5ry
*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*...##
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2D and 3D visibility in discrete geometry: an application to discrete geodesic paths

2004
*
Pattern Recognition Letters
*

Based on these definitions, we define

doi:10.1016/j.patrec.2003.12.002
fatcat:ig3trqlicveljj24zwssuqawzm
*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 ...##
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Lecture Notes on Gradient Flows and Optimal Transport
[article]

2010
*
arXiv
*
pre-print

We present a short overview on the strongest variational formulation for gradient flows of

arXiv:1009.3737v1
fatcat:bgiwzt537bbt3epk4z77ks27lu
*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. ...##
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Convex geodesic bicombings and hyperbolicity
[article]

2014
*
arXiv
*
pre-print

)

arXiv:1404.5051v1
fatcat:bmcljxyuwfeyfmeputwo64vfgy
*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. ...##
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Gradient flow structures for discrete porous medium equations
[article]

2012
*
arXiv
*
pre-print

We present a one-dimensional counterexample to

arXiv:1212.1129v1
fatcat:ydaua3rdibc3zmrufajgtzxxsq
*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 ...##
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A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature
[article]

2014
*
arXiv
*
pre-print

The main tool we use

arXiv:1403.4547v1
fatcat:7b6quis4pfb7xae63dkuoi5nt4
*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. ...##
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CAT(0) groups with specified boundary

2006
*
Algebraic and Geometric Topology
*

We specify exactly which groups can act geometrically on CAT(0)

doi:10.2140/agt.2006.6.633
fatcat:mnm3jljx4vgsfcril5dxpvy6ee
*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. ...##
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Discrete-time gradient flows and law of large numbers in Alexandrov spaces
[article]

2015
*
arXiv
*
pre-print

We develop the theory of

arXiv:1402.1629v2
fatcat:ekpipnhq2bbqbhi4a5bk4fiely
*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. ...##
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Energy of Twisted Harmonic Maps of Riemann Surfaces
[article]

2006
*
arXiv
*
pre-print

More generally, if ρ is a

arXiv:math/0506212v2
fatcat:a5bngted3jfobozhmz4ktpwmqq
*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 ...
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