Convex geometry is an area of mathematics between geometry, analysis and discrete mathematics. Classical discrete geometry is a close relative of convex geometry with strong ties to the geometry of numbers, a branch of number theory. Both areas have numerous relations to other fields of mathematics and its applications. While it is out of reach to describe on one or two dozen pages the main features of convex and discrete geometry, it is well possible to show the flavor of these areas by describing typical ideas, problems and results. This will be done in the following. In particular, we consider 1. Mixed volumes and the Brunn-Minkowski theorem, 2. Polar bodies in high dimensions, 3. Valuations, 4. Euler's polytope formula, 5. Lattice polytopes and lattice point enumerators, 6. Theorems of Minkowski and Minkowski-Hlawka, 7. Sums of moments, 8. Koebe's representation theorem.
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