## Inequalities: Selecta of Elliott H. LiebInequalities play a fundamental role in Functional Analysis and it is widely recognized that finding them, especially sharp estimates, is an art. E. H. Lieb has discovered a host of inequalities that are enormously useful in mathematics as well as in physics. His results are collected in this book which should become a standard source for further research. Together with the mathematical proofs the author also presents numerous applications to the calculus of variations and to many problems of quantum physics, in particular to atomic physics. |

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algebra assume ball boundary Brezis characteristic function classical compact compute concave function conjecture consider continuous convergence convex convex function Corollary Coulomb defined denote density matrix dimension dimensional E. H. Lieb eigenfunction eigenvalues ELLIOTT H energy entropy equality equation example exists fact fermion ferromagnetism finite fixed function f Gaussian functions Gaussian kernels Gaussian maximizers given Hamiltonian hence Hilbert space Hölder's inequality holds hypercontractivity implies integral Lebesgue measure Lemma Let f linear lower bound Lp spaces LP(R Math measure minimizing monotone nonnegative norm Note optimal orthogonal orthonormal particles Phys pointwise positive definite potential problem proof of Theorem prove quantum rearrangement inequality Remark replaced result right side satisfies Section sequence sharp constant singular Sobolev inequality solution spin subadditivity subspace Suppose supremum theory Thirring trace unique variables vector zero