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Displaying similar documents to “A positive solution to the Busemann-Petty problem in ℝ 4 .”

The determination of convex bodies from the size and shape of their projections and sections

Paul Goodey (2009)

Banach Center Publications

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We survey results concerning the extent to which information about a convex body's projections or sections determine that body. We will see that, if the body is known to be centrally symmetric, then it is determined by the size of its projections. However, without the symmetry condition, knowledge of the average shape of projections or sections often determines the body. Rather surprisingly, the dimension of the projections or sections plays a key role and exceptional cases do occur...

The skeleta of convex bodies

David G. Larman (2009)

Banach Center Publications

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The connectivity and measure theoretic properties of the skeleta of convex bodies in Euclidean space are discussed, together with some long standing problems and recent results.

A measure of axial symmetry of centrally symmetric convex bodies

Marek Lassak, Monika Nowicka (2010)

Colloquium Mathematicae

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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...