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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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