Surface bodies and \(p\)-affine surface area
The authors give a new characterization of Lutwak's (extended) \(p\)-affine surface area \[ O_p(K):=\int_{\partial K}\frac{\kappa(x)^{\frac {p}{n+p}}} {h(x)^{\frac{n(p-1)}{n+p}}}d\mu_{\partial K}(x)\quad (-\infty \leq p\leq+\infty,\;p\neq -n) \] where \(\kappa\) denotes the generalized Gauss curvature, \(h\) the support function and \(\mu_{\partial K}\) the usual surface measure of the boundary \(\partial K\) of a convex body \(K_{f,s}\) in \(R^n\) containing the origin in its interior. This characterization uses the so-called surface body \(K_{f,s}\) of \(K\) for a positive number \(s\) and a given probability density \(f\) on \(\partial K\), defined as the intersection of all half-spaces \(H^+\) such that \(\int_{\partial K\cap H^+}f(x)d\mu_{\partial K}(x)\leq s\). Then the main result in the paper is \[ d_n\lim_{s\to 0}\frac{\text{vol}_n(K)-\text{vol}_n(K_{f,s})} {s^{\frac{2}{n-1}}}= \int_{\partial K}\frac {\kappa^\frac{1}{n-1}(x)}{f^{\frac{2}{n-1}} (x)}d\mu_{\partial K} (x) \] for a suitable constant \(d_n\) depending only on the dimension \(n\) if additionally certain assumptions on \(f\) and \(\partial K\) are fulfilled. As a consequence, one gets for \(q:=\frac{n-p(n-2)}{p+1}\) and \(f=f_q: =\kappa^{\frac {q}{n+q}}(O_q(K)h^{\frac{n(q-1)}{n+q}})^{-1}\) the announced characterization of \(O_p(K)\): \[ d_n\lim_{s\to 0}\frac{\text{vol}_n(K)-\text{vol}_n(K_{f_q,s})}{\bigl(sO_q(K)\bigr)^{\frac{2}{n-1}}}=O_p(K). \]
scientific article; zbMATH DE number 2093005