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

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

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