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Discrete and Computational Geometry, WS1516 Exercise Sheet “8”: Lattices University of Bonn, Department of Computer Science I

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Discrete and Computational Geometry, WS1516 Exercise Sheet “8”: Lattices

University of Bonn, Department of Computer Science I

• Written solutions have to be prepared until Friday 22th of January, 12:00 pm.

• There is a letterbox in front of Room E.01 in the LBH builiding.

• You may work in groups of at most two participants.

Exercise 15: Integer Lattices (4 Points)

Prove: IfC ⊆Rd is convex, symmetric around the origin, bounded, and such that vol(C)> k·2d, thenC contains at least 2k lattice points.

Exercise 16: Determinant and Volume of Parallelepiped (4 points)

Let v1, . . . , vd be linearly independent vectors in Rd. Form a matrix A with v1, . . . , vd as rows. Prove that |detA| is equal to the volume of the paral- lelepiped {α1v1+α2v2+· · ·+αdvd |α1,· · ·, αd∈[0,1]}.

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