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 & Continuous Dynamical Systems - A, 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.  doi: doi:10.1051/cocv:2006002.  Google Scholar

[2]

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

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control,", Birkhäser, (2004).   Google Scholar

[4]

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

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Classics in Applied Mathematics, 5 (1990).   Google Scholar

[6]

F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).   Google Scholar

[7]

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

[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.   Google Scholar

[9]

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

[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.  doi: doi:10.1137/050630076.  Google Scholar

[11]

H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418.   Google Scholar

[12]

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

[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.  doi: doi:10.1016/j.na.2010.04.001.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften, 317 (1998).   Google Scholar

[18]

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

[19]

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

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.  doi: doi:10.1051/cocv:2006002.  Google Scholar

[2]

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

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control,", Birkhäser, (2004).   Google Scholar

[4]

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

[5]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Classics in Applied Mathematics, 5 (1990).   Google Scholar

[6]

F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998).   Google Scholar

[7]

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

[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.   Google Scholar

[9]

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

[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.  doi: doi:10.1137/050630076.  Google Scholar

[11]

H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418.   Google Scholar

[12]

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

[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.  doi: doi:10.1016/j.na.2010.04.001.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften, 317 (1998).   Google Scholar

[18]

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

[19]

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

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