This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.

Roughly speaking, a $d$-dimensional subset of $\mathbf R^n$ is minimizing if arbitrary deformations of it (in a suitable class) cannot decrease its $d$-dimensio

The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of ana

Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wea

This book deals with fractal geometries that have features similar to ones of ordinary Euclidean spaces, while at the same time being quite different from Eucli