A general geometric construction for affine surface area

• Autorzy: Elisabeth Werner
• Języki publikacji: EN

Abstrakty

EN

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].

Kategorie tematyczne

• 52A20: Convex sets in n dimensions (including convex hypersurfaces)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 132
• Numer: 3
• Strony: 227-238

Daty

wydano

1999

otrzymano

1997-06-30

otrzymano

1998-05-20

Twórcy

autor

Elisabeth Werner

• Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A.,
• Université de Lille 1, UFR de Mathématique, 59699 Villeneuve d'Ascq, France

Bibliografia

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• [Gr] P. Gruber, Aspects of approximation of convex bodies, in: Handbook of Convex Geometry, Vol. A, North-Holland, 1993, 321-345.
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• [L2] K. Leichtweiss, Über ein Formel Blaschkes zur Affinoberfläche, Studia Sci. Math. Hungar. 21 (1986), 453-474.
• [Lu] E. Lutwak, Extended affine surface area, Adv. Math. 85 (1991), 39-68.
• [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.
• [M-W] M. Meyer and E. Werner, The Santaló regions of a convex body, Trans. Amer. Math. Soc., to appear.
• [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.
• [S] C. Schütt, Floating body, illumination body, and polytopal approximation, preprint.
• [S-W] C. Schütt and E. Werner, The convex floating body, Math. Scand. 66 (1990), 275-290.
• [W] E. Werner, Illumination bodies and affine surface area, Studia Math. 110 (1994), 257-269.