Tr}\,}(-\Delta)\gamma{\geqslant}q^{-\frac2d}c_{\rm TF}\left(1-{\varepsilon}\right)\int_{{\mathbb{R}}^d}\rho_\gamma(x)^{1+\frac2d}dx
 -\frac{\kappa}{{\varepsilon}^{3+\frac4d}}\int_{{\mathbb{R}}^d}|\nabla\sqrt{\rho_\gamma}(x)|^2\,dx
 \label{eq:lower_bound_Nam}$$ for every ${\varepsilon}>0$ and some constant $\kappa=\kappa(d)$, in all space dimensions $d{\geqslant}1$.

We also recall the Hoffmann-Ostenhof inequality [@Hof-77] which states that $${{\rm Tr}\,}(-\Delta)\gamma{\geqslant}\int_{{\mathbb{R}}^d} \left|\nabla\sqrt{\rho_\gamma}(x)\right|^2\,dx
\label{eq:Hoffmann-Ostenhof}$$ and always imposes that $\sqrt{\rho}\in H^1({\mathbb{R}}^d)$. The inequality  does not require the fermionic constraint $0{\leqslant}\gamma{\leqslant}1$.

The Lieb-Oxford inequality [@Lieb-79; @LieOxf-80; @ChaHan-99; @LieSei-09] states that the total Coulomb energy is bounded from below by $$\begin{gathered}
\sum_{n=1}^{\infty}{{\rm Tr}\,}_{{\mathfrak{H}}^n}\Bigg(\sum_{1{\leqslant}j<k{\leqslant}n}\frac{1}{|x_j-x_k|}\Bigg)\Gamma_n\Bigg\}\\
{\geqslant}\frac12\int_{{\mathbb{R}}^3}\int_{{\mathbb{R}}^3}\frac{\rho(x)\rho(y)}{|x-y|}dx\,dy-1.64\int_{{\mathbb{R}}^3}\rho(x)^{\frac43}\,dx
\label{eq:Lieb-Oxford}\end{gathered}$$ where $\Gamma=\{\Gamma_n\}$ is a grand-canonical quantum state satisfying the conditions in . Inspired by [@BenBleLos-12], this bound was recently generalized in [@LewLie-15] to $$\begin{gathered}
\sum_{n=1}^{\infty}{{\rm Tr}\,}_{{\mathfrak{H}}^n}\Bigg(\sum_{1{\leqslant}j<k{\leqslant}n}\frac{1}{|x_j-x_k|}\Bigg)\Gamma_n\Bigg\}\\
{\geqslant}\frac12\int_{{\mathbb{R}}^3}\int_{{\mathbb{R}}^3}\frac{\rho(x)\rho(y)}{|x-y|}dx\,dy-\left(\frac35 \left(\frac{9\pi}{2}\right)^{\frac13}+{\varepsilon}\right)\int_{{\mathbb{R}}^3}\rho(x)^{\frac43}\,dx\\-\frac{0.001206}{{\varepsilon}^3}\int_{{\mathbb{R}}^3}|\nabla\rho(x)|\,dx
\label{eq:Lieb-Oxford-gradient}\end{gathered}$$ but the constant $(3/5) ({9\pi}/{2})^{1/3}\simeq 1.4508$ is not expected to be the optimal Lieb-Oxford constant.

Using  together with , we obtain the following.

\[cor:lower\_bound\_E\_rho\] We have $$E(\rho){\geqslant}q^{-\frac23}c_{\rm LT}\int_{{\mathbb{R}}^3} \rho(x)^{\frac53}\,dx-1.64\int_{{\mathbb{R}}^3}\rho(x)^{\frac43}\,dx
\label{eq:lower_bound_E}$$ for every $\rho{\geqslant}0$ such that $\sqrt{\rho}\in H^1({\mathbb{R}}^d)$.

The constants can be improved at the expense of adding gradient corrections, using  and .

Upper bound on the best kinetic energy {#sec:kinetic}
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Let us recall that the lowest possible kinetic energy of a fixed density $\rho\in L^1({\mathbb{R}}^d,{\mathbb{R}}_+)$ with $\sqrt\rho\in H^1({\mathbb{R}}^d)$ reads $$T(\rho):=\min_{\substack{0{\leqslant}\gamma=\gamma^*{\leqslant}1\\ \rho_\gamma=\rho}}{{\rm Tr}\,}(-\Delta)\gamma.$$ In [@LewLieSei-18], we have shown that for $\rho{\leqslant}1$, $$T(\rho){\leqslant}\int_{{\mathbb{R}}^d}|\nabla\sqrt\rho|^2+q^{-\frac2d}c_{\rm TF}\int_{{\mathbb{R}}^d}\rho$$ by using the trial state $$\gamma=\sqrt{\rho(x)}\,{{\ensuremath {\mathds 1} }}\left(-\Delta{\leqslant}\frac{d+2}d c_{\rm 