Uniform estimation of a convex body by a fixed-radius ball. (English) Zbl 1354.52009
Summary: The paper deals with a finite-dimensional problem of finding a uniform estimate (approximation in the Hausdorff metric) of a convex body by a fixed-radius ball in an arbitrary norm. The solution of the problem and its properties essentially depends on the radius value. We use the solutions of simpler problems of circumscribed and inscribed balls for the given convex body to divide the positive radius scale into ranges. Taking into account these ranges, we derive necessary and sufficient conditions for the solutions of the problem by means of convex analysis including properties of distant functions (to the nearest and most distant points of the set) and their differential characteristics. Also, under additional assumptions concerning the estimated convex body or the used norm, conditions of solution uniqueness and variational properties of the solution are obtained for various ranges of radius values.
References:
| [1] | Bonnesen, T., Fenchel, W.: Teorie der konvexen Körper. Springer, Berlin (1974) · Zbl 0277.52001 · doi:10.1007/978-3-642-93014-0 |
| [2] | Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, New Jersey (1970) · Zbl 0193.18401 · doi:10.1515/9781400873173 |
| [3] | Demyanov, V.F., Vasiliev, L.V.: Nondifferentiable Optimization. Springer-Optimization Software, New York (1986) |
| [4] | Demyanov, V.F., Rubinov, A.M.: Foundation of Nonsmooth Analysis and Quasidifferential Calculus. Nauka, Moscow (1990). (in Russian) · Zbl 0728.49001 |
| [5] | Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt a/M (1995) · Zbl 0887.49014 |
| [6] | Pschenichnyi, B.N.: Convex Analysis and Extremal Problems. Nauka, Moscow (1980). (in Russian) · Zbl 0477.90034 |
| [7] | Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 |
| [8] | Polovinkin, E.S., Balashov, M.V.: Elements of Convex and Strongly Convex Analysis. Fizmatlit, Moscow (2004). (in Russian) · Zbl 1181.26028 |
| [9] | Ivanov, G.E.: Weak Convex Set and Function. Theory and Applications. Fizmatlit, Moscow (2006). (in Russian) · Zbl 1171.26006 |
| [10] | Nikol’skii, M.S., Silin, D.B.: On the best approximation of a convex compact set by elements of addial. Proc. Steklov Inst. Math. 211, 306-321 (1995) · Zbl 0880.41031 |
| [11] | Dudov, S.I., Zlatorunskaya, I.V.: Best approximation of compact set by a ball in an arbitrary norm. Adv. Math. Res. 2, 81-114 (2003) · Zbl 1069.41020 |
| [12] | Dudov, S.I.: Relation between several problems of estimating convex compacta by balls. Sb. Math. 198(1), 39-53 (2007) · Zbl 1156.52005 · doi:10.1070/SM2007v198n01ABEH003828 |
| [13] | Dudov, S.I., Mescheraykova, E.A.: Method for finding an approximate solution of asphricity problem for a convex body. Comput. Math. Math. Phys. 53(10), 1483-1493 (2013) · Zbl 1299.41059 · doi:10.1134/S0965542513100059 |
| [14] | Dudov, S.I., Meshcheryakova, E.A.: On asphericity of convex body. Russ. Math. 2, 45-58 (2015) · Zbl 1321.52005 |
| [15] | Dudov, S.I.: Subdifferentiability and superdifferentiability of distance function. Math. Notes 61(4), 440-450 (1997) · Zbl 0913.90242 · doi:10.1007/BF02354988 |
| [16] | Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4(1), 79-97 (1979) · Zbl 0409.90086 · doi:10.1287/moor.4.1.79 |
| [17] | Gorokhovik, V.V.: Convex and Nonsmooth Problems of Vector Optimization. Nauvuka i Tekhnika, Minsk (1990) · Zbl 0765.90079 |
| [18] | Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley Interscience, New York (1983) · Zbl 1115.47049 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
