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3 Studia Mathematica

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A general geometric construction for affine surface area

100%

Werner E.

nr 3

227-238

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].

Illumination bodies and affine surface area

100%

Werner E.

nr 3

257-269

We show that the affine surface area as(∂K) of a convex body K in $ℝ^{n}$ can be computed as $as(∂K) = lim_{δ→0} d_{n} (vol_{n}(K^{δ}) - vol_{n}(K))/(δ^{2/(n+1)})$ where $d_{n}$ is a constant and $K^{δ}$ is the illumination body.

Approximation of the Euclidean ball by polytopes

51%

Ludwig M. , Schütt C. , Werner E.

Studia Mathematica

2006

tom 173

nr 1

1-18

There is a constant c such that for every n ∈ ℕ, there is an Nₙ so that for every N≥ Nₙ there is a polytope P in ℝⁿ with N vertices and $volₙ(B₂ⁿ△ P) ≤ c volₙ(B₂ⁿ)N^{-2/(n-1)}$ where B₂ⁿ denotes the Euclidean unit ball of dimension n.