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### Quasi-Banach Spaces which are Unique Predual.

Mathematische Annalen

### Illumination bodies and affine surface area

Studia Mathematica

We show that the affine surface area as(∂K) of a convex body K in ${ℝ}^{n}$ can be computed as $as\left(\partial K\right)=li{m}_{\delta \to 0}{d}_{n}\left(vo{l}_{n}\left({K}^{\delta }\right)-vo{l}_{n}\left(K\right)\right)/\left({\delta }^{2/\left(n+1\right)}\right)$ where ${d}_{n}$ is a constant and ${K}^{\delta }$ is the illumination body.

### A general geometric construction for affine surface area

Studia Mathematica

Let K be a convex body in ${ℝ}^{n}$ and B be the Euclidean unit ball in ${ℝ}^{n}$. We show that $li{m}_{t\to 0}\left(|K|-|{K}_{t}|\right)/\left(|B|-|{B}_{t}|\right)=as\left(K\right)/as\left(B\right)$, where as(K) respectively as(B) is the affine surface area of K respectively B and ${{K}_{t}}_{t\ge 0}$, ${{B}_{t}}_{t\ge 0}$ 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].

### Approximation of the Euclidean ball by polytopes

Studia Mathematica

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 $volₙ\left(B₂ⁿ△P\right)\le cvolₙ\left(B₂ⁿ\right){N}^{-2/\left(n-1\right)}$ where B₂ⁿ denotes the Euclidean unit ball of dimension n.

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