A general geometric construction for affine surface area

Elisabeth Werner

Studia Mathematica (1999)

  • Volume: 132, Issue: 3, page 227-238
  • ISSN: 0039-3223

Abstract

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Let K be a convex body in ℝ n and B be the Euclidean unit ball in ℝ n . We show that l i m t → 0 ( | K | - | K t | ) / ( | B | - | B t | ) = a s ( K ) / a s ( B ) , where as(K) respectively as(B) is the affine surface area of K respectively B and K t t ≥ 0 , B t t ≥ 0 are general families of convex bodies constructed from K,B satisfying certain conditions. As a corollary we get results obtained in [M-W], [Schm], [S-W] and [W].

How to cite

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Werner, Elisabeth. "A general geometric construction for affine surface area." Studia Mathematica 132.3 (1999): 227-238. <http://eudml.org/doc/216596>.

@article{Werner1999,
abstract = {Let K be a convex body in $ℝ^n$ and B be the Euclidean unit ball in $ℝ^n$. We show that $lim_\{t→ 0\} (|K| -|K_t|)/(|B| - |B_t|) = as(K)/as(B)$, where as(K) respectively as(B) is the affine surface area of K respectively B and $\{K_t\}_\{t≥0\}$, $\{B_t\}_\{t≥0\}$ are general families of convex bodies constructed from K,B satisfying certain conditions. As a corollary we get results obtained in [M-W], [Schm], [S-W] and [W].},
author = {Werner, Elisabeth},
journal = {Studia Mathematica},
keywords = {affine surface area; convex floating body; convolution body; illumination body; Santaló region},
language = {eng},
number = {3},
pages = {227-238},
title = {A general geometric construction for affine surface area},
url = {http://eudml.org/doc/216596},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Werner, Elisabeth
TI - A general geometric construction for affine surface area
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 3
SP - 227
EP - 238
AB - Let K be a convex body in $ℝ^n$ and B be the Euclidean unit ball in $ℝ^n$. We show that $lim_{t→ 0} (|K| -|K_t|)/(|B| - |B_t|) = as(K)/as(B)$, where as(K) respectively as(B) is the affine surface area of K respectively B and ${K_t}_{t≥0}$, ${B_t}_{t≥0}$ are general families of convex bodies constructed from K,B satisfying certain conditions. As a corollary we get results obtained in [M-W], [Schm], [S-W] and [W].
LA - eng
KW - affine surface area; convex floating body; convolution body; illumination body; Santaló region
UR - http://eudml.org/doc/216596
ER -

References

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  1. [B] W. Blaschke, Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie, Springer, 1923. Zbl49.0499.01
  2. [Gr] P. Gruber, Aspects of approximation of convex bodies, in: Handbook of Convex Geometry, Vol. A, North-Holland, 1993, 321-345. 
  3. [K] K. Kiener, Extremalität von Ellipsoiden und die Faltungsungleichung von Sobolev, Arch. Math. (Basel) 46 (1986), 162-168. Zbl0563.52009
  4. [L1] K. Leichtweiss, Zur Affinoberfläche konvexer Körper, Manuscripta Math. 56 (1986), 429-464. 
  5. [L2] K. Leichtweiss, Über ein Formel Blaschkes zur Affinoberfläche, Studia Sci. Math. Hungar. 21 (1986), 453-474. Zbl0561.53012
  6. [Lu] E. Lutwak, Extended affine surface area, Adv. Math. 85 (1991), 39-68. Zbl0727.53016
  7. [Lu-O] E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geometry 41 (1995), 227-246. Zbl0867.52003
  8. [M-W] M. Meyer and E. Werner, The Santaló regions of a convex body, Trans. Amer. Math. Soc., to appear. Zbl0917.52004
  9. [Schm] M. Schmuckenschläger, The distribution function of the convolution square of a convex symmetric body in ℝ n , Israel J. Math. 78 (1992), 309-334. Zbl0774.52004
  10. [S] C. Schütt, Floating body, illumination body, and polytopal approximation, preprint. Zbl0884.52007
  11. [S-W] C. Schütt and E. Werner, The convex floating body, Math. Scand. 66 (1990), 275-290. Zbl0739.52008
  12. [W] E. Werner, Illumination bodies and affine surface area, Studia Math. 110 (1994), 257-269. Zbl0813.52007

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