Do we have enough examples of Convex Bodies? Is diversity of our standard examples enough to understand Convexity? In the talk we demonstrate many different constructions which are analogous to constructions of irrational numbers from rationals. We show, following Il. Molchanov, that the solutions of "quadratic" equations like Z^o = Z + K always exists (where Z^o is the polar body of Z; Z and K are convex compact bodies containing 0 in the interior). Then we show how the geometric mean may be defined for any convex compact bodies K and T (containing 0 into their interior). We also construct K^a for any centrally symmetric K and 0 < a < 1, and also Log K for K containing the euclidean ball D (and K = −K). Note, the power a cannot be above 1 in the definition of power! All these constructions may be considered also for the infinite dimensional setting, but this is outside the subject of the talk.

These results are joint with Liran Rotem.