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Reduced spherical polygons

Marek Lassak — 2015

Colloquium Mathematicae

For every hemisphere K supporting a spherically convex body C of the d-dimensional sphere S d we consider the width of C determined by K. By the thickness Δ(C) of C we mean the minimum of the widths of C over all supporting hemispheres K of C. A spherically convex body R S d is said to be reduced provided Δ(Z) < Δ(R) for every spherically convex body Z ⊂ R different from R. We characterize reduced spherical polygons on S². We show that every reduced spherical polygon is of thickness at most π/2. We...

A measure of axial symmetry of centrally symmetric convex bodies

Marek LassakMonika 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...

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