New higher-order equiaffine invariants
The authors introduce new higher-order equiaffine invariants related to the affine surface area \(\Omega_0(K)\) for convex bodies \(K\) in \(\mathbb{R}^n\). For the construction they use the convex floating bodies \(K_\delta\) and the illumination bodies \(K^\delta\) like in the case of the definition of \(\Omega_0(K)\) for general convex bodies \(K\). Simplest example is \[ D_1\Omega(K):= d_1(n)\cdot\lim_{t\to 0}\,{\Omega_0(K)- \Omega_0(K_t)\over t} \] with a suitable universal, only dimension depending constant \(d_1(n)\) which exists if the boundary \(\partial K\) of \(K\) is of class \(C^4\). Then (if \(K\) is additionally strictly convex) there exists the integral representation \[ D_1\Omega(K)= \int_{\partial K} L_1 dV \] (\(L_1\) affine mean curvature, \(dV\) affine surface area element). Furthermore some sharp isoperimetric inequalities are established between the new invariants. The paper ends by computing them for \(l^p\)-unit balls in dimension 2.
scientific article; zbMATH DE number 5600328
scientific article