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We show that $$\\mbox{lim}_{t\\rightarrow 0} \\frac{|K| -|K_t|}{|B| - |B_t|}= \\frac{as(K)}{as(B)},$$ where $as(K)$ respectively $as(B)$ is the affine surface area of $K$ respectively $B$ and $\\{K_t\\}_{t\\geq 0}$, $\\{B_t\\}_{t\\geq 0}$ are general families of convex bodies constructed from $K$, $B$ satifying certain conditions. 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