## 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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### Contents

I | 33 |

II | 43 |

III | 47 |

IV | 59 |

V | 63 |

VI | 67 |

VII | 81 |

VIII | 91 |

XXVIII | 345 |

XXIX | 359 |

XXX | 367 |

XXXI | 377 |

XXXII | 391 |

XXXIII | 403 |

XXXIV | 417 |

XXXV | 441 |

IX | 95 |

X | 101 |

XI | 109 |

XII | 113 |

XIII | 135 |

XIV | 141 |

XV | 147 |

XVI | 151 |

XVII | 171 |

XVIII | 191 |

XIX | 203 |

XX | 239 |

XXI | 243 |

XXII | 255 |

XXIII | 269 |

XXIV | 305 |

XXV | 313 |

XXVI | 317 |

XXVII | 329 |

XXXVI | 465 |

XXXVII | 479 |

XXXVIII | 483 |

XXXIX | 497 |

XL | 515 |

XLI | 523 |

XLII | 529 |

XLIII | 555 |

XLIV | 563 |

XLV | 581 |

XLVI | 595 |

XLVII | 625 |

XLVIII | 633 |

XLIX | 637 |

L | 641 |

LI | 663 |

LII | 679 |

LIII | 695 |

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### Common terms and phrases

algebra assume ball boundary BRASCAMP Brezis classical co-area coherent compact concave function conjecture consider continuous convergence convex function Corollary Coulomb defined denote density matrix derivative dimension dimensional E. H. Lieb eigenvalues electrons ELLIOTT H entropy equality equation example exists fact fermion ferromagnetism finite fixed Gaussian functions Gaussian kernels Gaussian maximizers given Hamiltonian hence Hilbert space Holder's inequality holds hypercontractivity implies integral Lemma linear log concave function lower bound Lp norms LP(R Math measure minimizing monotone nonnegative norm Note one-dimensional optimal orthogonal orthonormal paper particles Phys Physics potential problem proof of Theorem properties prove quantum mechanical rearrangement inequality Remark replaced result right side satisfies Section sequence sharp constant singular Sobolev inequality solution spin subadditivity subspace Suppose symmetric decreasing rearrangement theory trace trace class unique variables vector Young's inequality zero