The kinetic energy inequality is a Gagliardo-Nirenberg type inequality for orthonormal functions but with a gain of summability. This inequality is dual to the fundamental Lieb-Thiring inequality on the sum of negative eigenvalues, and it is a useful tool to prove the stability of matter. In this talk, we give a proof of existence of an extremizer for the kinetic energy inequality and derive the Euler-Lagrange equation that is an infinitely many coupled nonlinear elliptic equations. I will briefly explain the concentration-compactness principle (or the profile decomposition) for functions, and show its extension for operators, which is one of our new tools. This is a joint work with Soonsik Kwon and Haewon Yoon at KAIST.