Vector balancing and quantized approximations in Banach spaces

The talk focuses on upper bounds for the vector balancing constant of a compact set $M$ in a Banach space $X$. We estimate this constant in terms of the sum of the Kolmogorov widths $d_n(M)$. We then apply this result to the density problem of quantized approximations by the set $M$. Let $R(M)$ denote the set of all possible sums of elements of $M$. We will show that if the sum of $d_n(M)$ is finite, the density of $R(M)$ in the weak topology of $X$ implies its density in the norm