In 1998 the authors associated to a given compact convex body \(K\) in the space \(\mathbb{R}^n\) with inner points and the Santaló point at the origin and to a positive real number \(t\) the Santaló regions \[ S(K,t):= \left\{x\in \text{int}(K); {|K||K^x|\over v^2_n}\leq t\right\}= \left\{x \in\text{int} (K);\int_{K^0} {dy\over(1-\langle x,y \rangle)^{n+1}} \leq{v^2_n \over |K|} t\right\} \] where \(|K||K^x|\) denotes the volume product of \(K\) and its polar body \(K^x\) with respect to \(x\in\text{int} (K)\) and \(v_n\) is the volume of the unit ball [see M. Meyer and E. Werner, Trans. Am. Math. Soc. 350, No. 11, 4569-4591 (1998; Zbl 0917.52004)]. They proved the following geometric interpretation of the affine surface area \(O_1(K)\) of \(K\) (slightly modificated) \[ O_1(K)= C_n\lim_{t\to \infty} {|K|- |S(K,{|K|\over v_n^2}t) |\over t^{-{2 \over n+1}}} (C_n \text{const}>0). \] The purpose of the paper is to show a comparable geometric interpretation of Lutwak’s \(p\)-affine surface area \[ O_p(K) :=\int_{\partial K}\left( {\kappa(x) \over\langle x,N(x) \rangle^{(p-1)^{n\over p}}}\right)^{p\over n+p} d\mu_K(x)\quad (p>-n) \] \((\kappa\) Gauss curvature, \(N\) outer unit normal vector, \(\mu_K\) surface area element of \(\partial K)\)
1) by means of suitably generalized Santaló-bodies (theorem 6) and 2) by means of the convex floating bodies of \(K\) (theorem 8). This is realized in the case of a body \(K\) with boundary of class \({\mathcal C}^3_+\) resp. class \({\mathcal C}^2_+\). Hereby in both interpretations of \(O_p(K)\) especially the polar body \(K^0\) of \(K\) with respect to the origin is involved.