### A conjecture on convex polyhedra.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Denote by Kₘ the mirror image of a planar convex body K in a straight line m. It is easy to show that K*ₘ = conv(K ∪ Kₘ) is the smallest by inclusion convex body whose axis of symmetry is m and which contains K. The ratio axs(K) of the area of K to the minimum area of K*ₘ over all straight lines m is a measure of axial symmetry of K. We prove that axs(K) > 1/2√2 for every centrally symmetric convex body and that this estimate cannot be improved in general. We also give a formula for axs(P) for...

We show that whenever the $q$-dimensional Minkowski content of a subset $A\subset {\mathbb{R}}^{d}$ exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ${\mathbb{R}}^{d}$, $d\ge 3$.

Let U, V be two symmetric convex bodies in ${\mathbb{R}}^{n}$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors ${u}_{1},...,{u}_{n}\in U$ such that, for each choice of signs ${\epsilon}_{1},...,{\epsilon}_{n}=\pm 1$, one has ${\epsilon}_{1}{u}_{1}+...+{\epsilon}_{n}{u}_{n}\notin rV$ where $r={\left(2\pi {e}^{2}\right)}^{-1/2}{n}^{1/2}\left(\right|U|/{\left|V\right|)}^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $\left({u}_{n}\right)$ such that the series ${\sum}_{n=1}^{\infty}{\epsilon}_{n}{u}_{\pi \left(n\right)}$ is divergent for any choice of signs ${\epsilon}_{n}=\pm 1$ and any permutation π of indices.