A characterization of affine length and asymptotic approximation of convex discs

Abstract

It is shown that every equi-affine invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, area, and affine length. Asymptotic formulae for approximation of convex discs by polygons are derived, extending results of L. Fejes Tóth from smooth convex discs to general convex discs. 1991

A Characterization of Affine Surface Area

“…In the planar case, this theorem was proved in [19] and independently by Tibor O dor. Recently, he informed us that he has also obtained the above theorem.…”

Valuations on Polytopes Containing the Origin in Their Interiors

“…In [15] and in the joint paper with Reitzner [17], a classification of upper semicontinuous, equi-affine invariant valuations on the space of convex bodies is given. A functional on K d is called equi-affine invariant if it is invariant with respect to translations and SLðdÞ: These functionals are linear combinations of the Euler characteristic, volume, and affine surface area.…”

Valuations on Polytopes Containing the Origin in Their Interiors

“…In [15] and in the joint paper with Reitzner [17], a classification of upper semicontinuous, equi-affine invariant valuations on the space of convex bodies is given. A functional on K d is called equi-affine invariant if it is invariant with respect to translations and SL(d).…”