During the last few years there has been a systematic pursuit for sharp estimates 

of the energy components of atomic systems in terms of their single particle density. 

The common feature of these estimates is that they include corrections that 

depend on the gradient of the density. In this talk I will review these results. 

The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. 

I will also present some recent results concerning geometric estimates 

for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. 

These geometric estimates are a useful tool to estimate the numerical value of the constant of 

Nam's gradient correction term.