On coverings of Banach spaces and their subsets by hyperplanes

Given a Banach space we consider the $\sigma$-ideal of all its subsets which are covered by countably many hyperplanes and investigate standard cardinal characteristics as additivity, covering number, uniformity, cofinality. We determine their values for separable spaces, approximate them nonseparable spaces. The remaining questions reduce to deciding if following can be proved in ZFC (i.e. Zermelo-Fraenkel set theory with axiom choice) every $X$: (1) $X$ $\omega _1$-many hyperplanes; (2) All cardinalities less than $\operatorname {cf}([\operatorname {dens}(X)]^\omega )$ hyperplanes. prove spaces well-investigated classes that they consistent any possible size continuum. is related problem whether compact Hausdorff has small diagonal metrizable large cardinals.