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

Paul Goodey (2009)

Banach Center Publications

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 but appear to...

The Different Ways of Stabbing Disjoint Convex Sets.

M. Katchalski, T. Lewis, A. Liu (1992)

Discrete & computational geometry

The dimension of almost spherical sections of convex bodies

J. Lindenstrauss (1975/1976)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

The Dimension Print of Most Convex Surfaces.

Tudor Zamfirescu, G. Craciun (1997)

Monatshefte für Mathematik

The dissection of five-dimensional simplices into orthoschemes.

Tschirpke, Katrin (1994)

Beiträge zur Algebra und Geometrie

The distance between subdifferentials in the terms of functions

Libor Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

For convex continuous functions $f, g$ defined respectively in neighborhoods of points $x, y$ in a normed linear space, a formula for the distance between $\partial f (x)$ and $\partial g (y)$ in terms of $f, g$ (i.eẇithout using the dual) is proved. Some corollaries, like a new characterization of the subdifferential of a continuous convex function at a point, are given. This, together with a theorem from [4], implies a sufficient condition for a family of continuous convex functions on a barrelled normed linear space to be locally uniformly...

The distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous planar anisotropic STIT Tessellations

Christoph Thäle (2010)

Commentationes Mathematicae Universitatis Carolinae

A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented...

The distribution of unbounded random sets and the multivalued strong law of large numbers in nonreflexive Banach spaces.

Hess, Christian (1999)

Journal of Convex Analysis

The dual form of Knaster-Kuratowski-Mazurkiewicz principle in hyperconvex metric spaces and some applications

George Isac, George Xian-Zhi Yuan (1999)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we first establish the dual form of Knaster- Kuratowski-Mazurkiewicz principle which is a hyperconvex version of corresponding result due to Shih. Then Ky Fan type matching theorems for finitely closed and open covers are given. As applications, we establish some intersection theorems which are hyperconvex versions of corresponding results due to Alexandroff and Pasynkoff, Fan, Klee, Horvath and Lassonde. Then Ky Fan type best approximation theorem and Schauder-Tychonoff fixed point...