| RUS ENG |
| Full version |
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| PEOPLE |
| Dudov Sergey Ivanovitch |
| Professor |
| Doctor of physico-mathematical sciences (1997) |
The differential properties of the distance function in an arbitrary norm from a point to a set for the finite-dimensional case were studied:
a) necessary and sufficient conditions for its directional differentiability at a fixed point were obtained and the corresponding formula was found for directional derivative, provided that the directional differentiability is valid;
b) necessary and sufficient conditions for its subdiffentiability and superdifferentiability (in V. F. Demyanov–A. M. Rubinov sence) at a fixed point and corresponding formulae for subdifferential and superdifferential were found;
c) it was demonstrated how to find an upper convex approximation (u.s.a.) and lower concave approximation (l.c.a.) (in the sence of B. N. Pshenichnyi) of the distance function and how to construct an exhaustive family of u.c.a.s and lower exhaustive family of l.c.a.s.;
d) its formula of Penot subdifferential and outer estimate of Clarks subdifferential were obtained. The properties of the distance function and some other marginal functions were applied for outer and inner estimating and also for best approximating of a given convex compact set by the balls in an arbitrary norm.
The problem of the best approximation of a compact convex set by a ball with respect to an arbitrary norm in the Hausdorff metric corresponding to this norm was reduced to a convex programming problem and was studied by means of convex analysis. Necessary and sufficient condition of its solution was obtained and several properties of its solution were described. It is proved that the center of at least one ball of its best approximation lies in the approximated set; the conditions ensuring that the centers of all balls of the best approximation lie in this compact set and conditions for unique solubility were obtained; in addition several variational properties of the solutions were studied.
