Iterated means of convex bodies (Q1207687)

Let \(B_ 1\), \(B_ 2\) and \(K\) be convex bodies in \(\mathbb{R}^ n\) with 0 as an interior point and choose real numbers \(\lambda\), \(p\), such that \(0<\lambda<1\), \(p\geq 1\). Let \(F_ i(x)\) and \(H_ i(x)\) be, respectively, the distance (or gauge) function and the support function of \(B_ i\), \(i=1,2\). W. Firey defined the weighted \(p\)-mean, \(\lambda B_ 1+_ p(1-\lambda)B_ 2\), as the convex body with support function \((\lambda H_ 1(x)^ p+(1-\lambda)H_ 2(x)^ p)^{1/p}\) and the weighted \(p\)-dot mean by \[ \lambda B_ 1+_ p(1-\lambda)B_ 2=\{x:(\lambda F_ 1(x)^ p+(1-\lambda)F_ 2(x)^ p)^{1/p}\leq 1\}. \] Here \(K^ \wedge\) denotes the polar of \(K\); \(K^ \wedge=\{y:\;x\cdot y\leq 1\), \(x\in K\}\), where \(x\cdot y\) is the usual inner product on \(\mathbb{R}^ n\). Our main result is the following theorem. Theorem. Choose \(\lambda,p\) and \(K\) as above. Let \(K_ 0=K\) and for each \(i=0,1,2,\dots,\) let \(\oplus_ p\) denote a \(p\)-mean or a \(p\)-dot mean (the choice may depend on \(i)\). Let \(K_{i+1}=\lambda K_ i\oplus_ p(1-\lambda)K_ i^ \wedge\), \(i=0,1,2,\dots\) Then, as \(i\to\infty\), \(\lim K_ i\) is the unit ball in \(\mathbb{R}^ n\). When \(n=1\), this is related to well known examples in \(\mathbb{R}\) such as \(\lim_{i\to\infty}x_ i=1\), when \(x_ 0>1\) and \(x_{i+1}=(1/2)x_ i+(1/2)x_ i^{-1}\). An estimate on the rate of convergence is given, with better estimates given for the simple case of ellipsoids \(K\) centered at the origin with \(p=2\).