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A general geometric construction for affine surface area. (English) Zbl 0927.52008
It is shown that the (equivalent) extensions $$as(K)$$ of affine surface area to an arbitrary convex body $$K$$ in $$\mathbb{R}^n$$ by C. Schütt and E. Werner [Math. Scand. 66, No. 2, 275-290 (1990; Zbl 0739.52008)] resp. M. Schmuckenschläger [Isr. J. Math. 78, No. 2-3, 309-334 (1992; Zbl 0774.52004)] resp. E. Werner [Stud. Math. 110, No. 3, 257-269 (1994; Zbl 0813.52007)] resp. M. Meyer and E. Werner [Trans. Am. Math. Soc. 350, No. 11, 4569-4591 (1998; Zbl 0917.52004)] have a common feature: These mathematicians construct specific families $$\{K_t\}$$ $$(t\geq 0)$$ of convex bodies from $$K$$ (namely the convex floating bodies resp. the convolution bodies resp. the illumination bodies resp. the Santaló regions) and get as main result for $$as(K)$$ $\lim_{t\to+0} {| K|-| K_t|\over| B|-| B_t|}= {as(K)\over as(B)}\quad (B\text{ unit ball in }\mathbb{R}^n,\;as(B)=n| B|).$ Now, the author proves like in the first of the quoted papers the validity of the formula above for a general affine equivariant and monotone family $$\{K_t\}$$ ($$t\geq 0$$) with $$K_0= K$$ satisfying $\lim_{t\to+0} {| B|-| B_t|\over t^{{2\over n+1}}}= \pm c_n$ and some additional conditions.

MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
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