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A measure of axial symmetry of centrally symmetric convex bodies

Marek Lassak, Monika Nowicka (2010)

Colloquium Mathematicae

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...

Almost sure asymptotic behaviour of the r -neighbourhood surface area of Brownian paths

Ondřej Honzl, Jan Rataj (2012)

Czechoslovak Mathematical Journal

We show that whenever the q -dimensional Minkowski content of a subset A 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 d , d 3 .

Balancing vectors and convex bodies

Wojciech Banaszczyk (1993)

Studia Mathematica

Let U, V be two symmetric convex bodies in n and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors u 1 , . . . , u n U such that, for each choice of signs ε 1 , . . . , ε n = ± 1 , one has ε 1 u 1 + . . . + ε n u n r V where r = ( 2 π e 2 ) - 1 / 2 n 1 / 2 ( | U | / | V | ) 1 / n . Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence ( u n ) such that the series n = 1 ε n u π ( n ) is divergent for any choice of signs ε n = ± 1 and any permutation π of indices.

Complete systems of inequalities.

Hernández Cifre, María A., Salinas, Guillermo, Segura Gomis, Salvador (2001)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

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