Engineering topology optimization is a technique to minimize the mass of a structure while maintaining or even increasing its robustness in certain lifecycle applications. The presentation will show the technical foundation based on the finite element method with an embedded gradient based mathematical optimizer. Characteristic application solutions utilizing the method and the intriguing connection to additive manufacturing (3D printing) will also be discussed.

 In 2006 Carbery raised a question about an improvement on the naïve norm inequality 
(||f+g||_p)^p ≤ 2^(p-1)((||f||_p)^p + (||g||_p)^p) for two functions in Lp of any measure space. When f=g this is an equality, but when the supports of f and g are disjoint the factor 2^(p-1) is not needed. Carbery’s question concerns a proposed interpolation between the two situations for p>2. The interpolation parameter measuring the overlap is ||fg||_(p/2). We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all p (joint work with E. A. Carlen, R. L. Frank, and  E. H. Lieb).