A general geometric construction for affine surface area

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