David Hilbert’s (1862–1943) so-called Zahlbericht of 1897, which appears here for the first time in English, was the principal textbook on algebraic number theory for a period of at least thirty years after its appearance. Emil Artin, Helmut Hasse, Erich Hecke, Hermann Weyl and many others learned their number theory from this book. Even beyond this immediate impact Hilbert’s Zahlbericht has se...

Let A, B ⊆ R be finite, nonempty subsets, let s ≥ 2 be an integer, and let h1(A,B) denote the minimal number t such that there exist 2t (not necessarily distinct) parallel lines, l1, . . . , lt, l ′ 1, . . . , l ′ t , with A ⊆ ⋃t i=1 li and B ⊆ ⋃t i=1 l ′ i . Suppose h1(A,B) ≥ s. Then we show that: (a) if ||A| − |B|| ≤ s and |A|+ |B| ≥ 4s − 6s+ 3, then |A+B| ≥ (2− 1 s )(|A| + |B|)− 2s+ 1; (b) i...

In this article, we introduce the concept of general Lp-mixed chord integral difference star bodies. Further, establish Brunn–Minkowski type, Aleksandrov–Fenchel type and cyclic inequalities for difference.

Two consequences of the stability version of the one dimensional Prékopa-Leindler inequality are presented. One is the stability version of the Blaschke-Santaló inequality, and the other is a stability version of the Prékopa-Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn-Minkowski inequality is also used in an essential way. 1 The prob...

Consider convex plane domains D(t) = (1− t)D0 + tD1, 0 ≤ t ≤ 1. We first prove that the 1/4-power of the polar moment of inertia about the centroid of D(t) is concave in t. From this we deduce some isoperimetric inequalities.