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Search: MSC category 52A20 ( Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45] )

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1. CJM Online first

Alfonseca, M. Angeles; Kim, Jaegil
On the Local Convexity of Intersection Bodies of Revolution
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

Keywords:convex bodies, intersection bodies of star bodies, Busemann's theorem, local convexity
Categories:52A20, 52A38, 44A12

2. CJM 2009 (vol 61 pp. 299)

Dawson, Robert J. MacG.; Moszy\'{n}ska, Maria
\v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$
A set in a metric space is called a \emph{\v{C}eby\v{s}ev set} if it has a unique ``nearest neighbour'' to each point of the space. In this paper we generalize this notion, defining a set to be \emph{\v{C}eby\v{s}ev relative to} another set if every point in the second set has a unique ``nearest neighbour'' in the first. We are interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of \v{C}eby\v{s}ev and relatively \v{C}eby\v{s}ev sets in various hyperspaces. In particular, we show that certain nested families of sets are \v{C}eby\v{s}ev. As these families are characterized purely in terms of containment, without reference to the semi-linear structure of the underlying metric space, their properties differ markedly from those of known \v{C}eby\v{s}ev sets.

Keywords:convex body, strictly convex set, \v{C}eby\v{s}ev set, relative \v{C}eby\v{s}ev set, nested family, strongly nested family, family of translates
Categories:41A52, 52A20

3. CJM 2007 (vol 59 pp. 1029)

Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M.
The Geometry of $L_0$
Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations --- linear transformations, closure in the radial metric, and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$ We prove that in dimension $3$ this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions $4$ and higher. We introduce the concept of embedding of a normed space in $L_0$ that naturally extends the corresponding properties of $L_p$-spaces with $p\ne0$, and show that the procedure described above gives exactly the unit balls of subspaces of $L_0$ in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in $L_0$, and prove several facts confirming the place of $L_0$ in the scale of $L_p$-spaces.

Categories:52A20, 52A21, 46B20

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