Weighted geometric means of convex bodies

Given two convex bodies K and T and a number 0 < λ < 1, we present two constructions for the geometric mean of K and T with weights 1 − λ and λ. If K and T are the unit balls of two norms, one may think of the weighted geometric means as the unit balls of a “geometric interpolation” between the two normed spaces. Our first construction is explicit, but lacks the so called “duality prop-erty” – a natural property one expects from the geometric mean. The second construction has this property, but is based on the choice of an ultrafilter. Both definitions extend previous constructions by the authors for the power K^λ, and the results proved here also prove some previously announced results.