Sergei Zuyev: Measures everywhere!

Branches of science using measures are abound: length, volume, mass, electric charge, probability, point processes are all examples of measures. Looking at the set of measures as a convex cone in a Banach space we derive specific first and second order necessary conditions for extrema of functionals of measures. We discuss applications in numeric approximation, probability, statistics and optimal experimental design. The obtained explicit formula for the gradient allows us to develop new steepest descent algorithms efficiency of which will be shown on numeric examples.