Convolution bodies and their limiting behavior (Q1355296)

Let \(K\) and \(L\) denote convex symmetric bodies in \(\mathbb{R}^n\) and define their \(\delta\)-convolution body by \[ C(\delta;K,L):=\{x\in\mathbb{R}^n\mid\text{vol}_n(K\cap(x+L))\geq \delta\text{ vol}_n(K\cap L)\}\text{ with }0<\delta<1 \] The author examines the limiting convolution body of \(K\) and \(L\) defined by \(C(K,L):=\lim_{\delta\uparrow 1}(1-\delta)^{-\alpha}C(\delta;K,L)\) where \(\alpha\) is chosen such that the limit exists. He shows (Theorem 3.2) that \(C(K,L)\) exists with \(\alpha=1\) if \(L\) is ``well fitted'' in \(K\) and if the intersection of the boundaries of \(K\) and \(L\) is ``not too small''. In the case \(K=L\) the limiting convolution body turns out to be the polar of the projection body of \(K\) (as already shown by \textit{M. Schmuckenschläger} [Isr. J. Math. 78, No. 2-3, 309-334 (1992; Zbl 0774.52004)]. The author also shows (Theorem 3.9) that \(C(K,L)\) exists if \(\alpha=1/2\) and \(K\) and \(L\) are ``generic in relative position''. In that case \(C(K,L)\) is an ellipsoid. Examples show that, in other cases, \(C(K,L)\) can be very different from an ellipsoid.