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Volume approximation of convex bodies by polytopes - a constructive method

100%

Gordon Y. , Meyer M. , Reisner S.

Studia Mathematica

1994

tom 111

nr 1

81-95

Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in $ℝ^d$, so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than $f(d)/n^{2/(d-1)}$.

Determination of convex bodies by infinity-chord functions

100%

Sójka G.

Demonstratio Mathematica

2010

tom Vol. 43, nr 1

187-202

In 1998 A. Soranzo introduced the notions of +infinity - and - infinity-chord functions (see [16]). In this paper we give an answer to the question when a convex body is determined by the values of -infinity-chord functions at chosen internal points. We also give some partial results regarding + infinity chord functions.

Monotone Valuations on the Space of Convex Functions

88%

Cavallina L. , Colesanti A.

Analysis and Geometry in Metric Spaces

2015

tom 3

nr 1

We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.