Radchenko D. High-dimensional shape-preserving approximation

The topic of this Candidate's thesis is the study of approximation by mappings with nonnegative Jacobian, comonotone approximation on the interval, and new bounds for Whitney constants of high-dimensional cubes. We study conditions for when a mapping can be approximated by mappings with nonnegative Jacobian, find all exponents for which the degree of comonotone approximation has the same power decay as the unconstrained degree, and improve bounds for Whitney constants for linear appoximation on cubes. We also prove that the Riesz energy function of point sets on spheres does not have local maxima.