convex body

On an Inner Estimate of a Convex Body by the Lebesgue Set of Convex Differentiable Function

Authors: Dudov S. I., Abramova V. V.

A finite-dimentional problem of embedding the largest by the inclusion of lower Lebesgue set of given convex function f(x) in a given convex body D ⊂ R p is considered. This problem is the generalization of the problem of inscribed ball (function f(x) is some norm, and the Lebesgue sets are the corresponding balls). The function f(x) must be differentiable on R p possibly expending the point 0 p and 0 p is the uniqueness point of minimum. Mathematical formalization of this problem is proposed in the form of finding maximin of a function of the difference of arguments.

On Functional Stability of the Solution for the Problem of Convex Body Best Approximating by a Ball with Fixed Radius

Authors: Dudov S. I., Osipcev M. A.

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 is considered. It is known that this problem can be reduced to a linear programming problem in the case, when the convex body and the norm ball are polytops. Therefore, we prove the functional stability of the optimal value of the objective function with respect to accuracy of the given convex body and accuracy of the unit ball for the norm used. The stability rating is derived.