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Generalized exterior sphere conditions and $\varphi$-convexity

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  • 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.
    Mathematics Subject Classification: Primary: 49J52; Secondary: 52A20.

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  • [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 $\R^n$, 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.

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