Search results

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

 Expand all        Collapse all Results 1 - 2 of 2

1. 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 translatesCategories:41A52, 52A20

2. CJM 2007 (vol 59 pp. 1029)

 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