April  2011, 29(2): 615-622. doi: 10.3934/dcds.2011.29.615

Generalized exterior sphere conditions and $\varphi$-convexity

1. 

Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos

2. 

Department of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada

3. 

Department of Computer Science and Mathematics, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos, Lebanon

Received  August 2009 Revised  March 2010 Published  October 2010

We consider sets $S\subset\R^n$ satisfying a certain exterior sphere condition, and it is shown that under wedgedness of $S$, it coincides with $\varphi$-convexity. We also offer related improvements concerning the union of uniform closed balls conjecture.
Citation: Chadi Nour, Ron J. Stern, Jean Takche. Generalized exterior sphere conditions and $\varphi$-convexity. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 615-622. doi: 10.3934/dcds.2011.29.615
References:
[1]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370. doi: doi:10.1051/cocv:2006002.

[2]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298. doi: doi:10.1007/BF01189393.

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004.

[4]

A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157. doi: doi:10.1016/0022-0396(88)90132-5.

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.

[6]

F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.

[7]

F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: doi:10.1137/S036301290240453X.

[8]

F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144.

[9]

G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31. doi: doi:10.1007/s00526-005-0352-7.

[10]

G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: doi:10.1137/050630076.

[11]

H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.

[12]

C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514.

[13]

C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589. doi: doi:10.1016/j.na.2010.04.001.

[14]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: doi:10.1090/S0002-9947-96-01544-9.

[15]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: doi:10.1090/S0002-9947-00-02550-2.

[16]

R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{n}$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154. doi: doi:10.1016/0362-546X(79)90044-0.

[17]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998.

[18]

A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259. doi: doi:10.1137/0804006.

[19]

C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774. doi: doi:10.3934/cpaa.2004.3.757.

show all references

References:
[1]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370. doi: doi:10.1051/cocv:2006002.

[2]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298. doi: doi:10.1007/BF01189393.

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004.

[4]

A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157. doi: doi:10.1016/0022-0396(88)90132-5.

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.

[6]

F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.

[7]

F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: doi:10.1137/S036301290240453X.

[8]

F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144.

[9]

G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31. doi: doi:10.1007/s00526-005-0352-7.

[10]

G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: doi:10.1137/050630076.

[11]

H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.

[12]

C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514.

[13]

C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589. doi: doi:10.1016/j.na.2010.04.001.

[14]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: doi:10.1090/S0002-9947-96-01544-9.

[15]

R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: doi:10.1090/S0002-9947-00-02550-2.

[16]

R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbb{R}^{n}$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154. doi: doi:10.1016/0362-546X(79)90044-0.

[17]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998.

[18]

A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259. doi: doi:10.1137/0804006.

[19]

C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774. doi: doi:10.3934/cpaa.2004.3.757.

[1]

Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389

[2]

Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295

[3]

Vladimir F. Demyanov, Julia A. Ryabova. Exhausters, coexhausters and converters in nonsmooth analysis. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1273-1292. doi: 10.3934/dcds.2011.31.1273

[4]

Chjan C. Lim, Da Zhu. Variational analysis of energy-enstrophy theories on the sphere. Conference Publications, 2005, 2005 (Special) : 611-620. doi: 10.3934/proc.2005.2005.611

[5]

Carlo Sinestrari. Semiconcavity of the value function for exit time problems with nonsmooth target. Communications on Pure and Applied Analysis, 2004, 3 (4) : 757-774. doi: 10.3934/cpaa.2004.3.757

[6]

Guoyong Gu, Junfeng Yang. A unified and tight linear convergence analysis of the relaxed proximal point algorithm. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022107

[7]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789

[8]

P. Cerejeiras, M. Ferreira, U. Kähler, F. Sommen. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Communications on Pure and Applied Analysis, 2007, 6 (3) : 619-641. doi: 10.3934/cpaa.2007.6.619

[9]

Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559

[10]

Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79

[11]

Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431

[12]

Qi An, Chuncheng Wang, Hao Wang. Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5845-5868. doi: 10.3934/dcds.2020249

[13]

Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048

[14]

Inbo Sim. On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition. Communications on Pure and Applied Analysis, 2008, 7 (4) : 905-923. doi: 10.3934/cpaa.2008.7.905

[15]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045

[16]

Jie Shen, Jian Lv, Fang-Fang Guo, Ya-Li Gao, Rui Zhao. A new proximal chebychev center cutting plane algorithm for nonsmooth optimization and its convergence. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1143-1155. doi: 10.3934/jimo.2018003

[17]

Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2611-2631. doi: 10.3934/jimo.2021084

[18]

Marcio V. Ferreira, Gustavo Alberto Perla Menzala. Uniform stabilization of an electromagnetic-elasticity problem in exterior domains. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 719-746. doi: 10.3934/dcds.2007.18.719

[19]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103

[20]

Wei Sun. On uniform estimate of complex elliptic equations on closed Hermitian manifolds. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1553-1570. doi: 10.3934/cpaa.2017074

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]