P. A. Borodin

An example of nonexistence of a steiner point in a Banach space

For each n = 3, 4, …, we construct an example of a Banach space X and elements x1, …, xn in this space such that X does not have any element with the minimal sum of distances to the elements xk.

Estimates of the distances to direct lines and rays from the poles of simplest fractions bounded in the norm of L p on these sets

For each p > 1, we obtain a lower bound for the distances to the real axis from the poles of simplest fractions (i.e., logarithmic derivatives of polynomials) bounded by 1 in the norm of Lp on this axis; this estimate improves the first estimate of such kind derived by Danchenko in 1994. For p = 2, the estimate turns out to be sharp. Similar estimates are obtained for the distances from the poles of simplest fractions to the vertices of angles and rays.

The Banach--Mazur Theorem for Spaces with Asymmetric Norm

AbstractWe establish an analog of the Banach—Mazur theorem for real separable linear spaces with asymmetric norm: every such space can be linearly and isometrically embedded in the space of continuous functions f on the interval [0,1] equipped with the asymmetric norm

The linearity coefficient of metric projections onto a Chebyshev subspace

||f|

An example of non-approximatively-compact existence set with finite-valued metric projection

. This assertion is used to obtain nontrivial representations of an arbitrary convex closed body

Quantitative expressions for the connectedness of sets in ℝ n

M \subset \mathbb{R}^n