For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to $\mathfrak{c}$.

In this paper we give a lower bound on the waist of the unit sphere of a uniformly convex normed space by using the localization technique in codimension greater than one and a strong version of the Borsuk–Ulam theorem. The tools used in this paper follow ideas of Gromov in [Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178–215] and we also include an independent proof of our main theorem which does not rely on Gromov’s waist of the sphere. Our waist inequality in codimension one recovers a version of the Gromov–Milman inequality in [Generalisation of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282].

Given a centrally symmetric convex body $B$ in ${{\mathbb{E}}^{d}}$, we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$. Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$. The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$, where $\alpha \left( B \right)>0$. As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.