exercise sheet 3 - tu berlin€¦ · exercise 3.2 let 0 ˆ 2ln be a sublattice, let a 2 0 be...
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TU BerlinMartin Henk
Discrete Geometry III
SS 2015
Exercise sheet 3
Discussion: Wednesday, 06.05.2015.
Exercise 3.1 Let Λ ∈ Ln and let Ln−1 ∈ L(n−1,Λ). Show that there existsa primitive vector b? ∈ Λ∗ with
i) Ln−1 = {x ∈ Rn : 〈b?,x〉 = 0}.
ii) det(Λ ∩ Ln−1) = det Λ · |b∗|.
Exercise 3.2 Let Λ0 ⊂ Λ ∈ Ln be a sublattice, let a ∈ Λ0 be primitivew.r.t. Λ0, and let α ∈ N such that 1
αa is primitive w.r.t. Λ. Show that α isa divisor of the index [Λ : Λ0].
Exercise 3.3 Let k ∈ N≥1, K ∈ Kn0 and let Λ ∈ Ln with vol (K) ≥k 2n det Λ. Then
# (K ∩ Λ) ≥ 2k + 1.
Exercise 3.4 Let P ⊂ Rn be a lattice polytope having k interior latticepoints of the integral lattice Zn. Show that
vol (P ) ≥ nk + 1
n!.
Is this inequality best possible?
TU BerlinMartin Henk
Discrete Geometry III
SS 2015
Exercise sheet 2
Discussion: Wednesday, 29.04.2015.
Exercise 2.1 Let A = (a1, . . . , an) be linearly independent lattice vectors ofa lattice Λ ∈ Ln. Show that there exists a basis B = (b1, . . . , bn) of Λ andan upper triangular matrix Z = (zi,j) ∈ Zn×n such that A = B Z and for1 ≤ k ≤ n it holds
0 ≤ zi,k < zk,k, 1 ≤ i ≤ k − 1. (1)
In addition, (if weather permits) show that for given A such a basis B isuniquely determined.
Exercise 2.2 Let Λ = Zn. Give an example of a k-dimensional subspace Lsuch that Λ|L⊥ is not a lattice.
Exercise 2.3 Let Λ ∈ L2 and let a1,a2 ∈ Λ be linearly independent. Then
a1, a2 is basis of Λ⇔ conv {0,a1,a2} ∩ Λ = {0,a1,a2}.
Exercise 2.4 Let a1, . . . ,an ∈ Λ ∈ Ln be linearly independent, PA ={∑n
i=1 ρiai : 0 ≤ ρi < 1}, and let t ∈ Rn. Prove or disprove that
#((t + PA) ∩ Λ) = #(PA ∩ Λ).
Exercise 2.5 Let P ∈ Pn be a polytope containing n+1 affinely independentintegral points of the lattice Zn. Show that
#(P ∩ Zn) ≤ n!vol (P ) + n.
Is the inequality best possible?
TU BerlinMartin Henk
Discrete Geometry III
SS 2015
Exercise sheet 1
Discussion: Wednesday, 22.04.2015.
Exercise 1.1 Let X ⊆ Rn, t ∈ Rn and let 0 ∈ intX ∩ int (t + X). Showthat
(t +X)? =
{1
1 + 〈t,y〉y : y ∈ X?
}.
Exercise 1.2 Show that the number of simplices in any dissection of thecube B∞n into simplices having vertices of B∞n can not be smaller than2n n!/(n+ 1)(n+1)/2.
Exercise 1.3 Let K ∈ Kno with intK ∩ Zn = 0, i.e., 0 is the only latticepoint in the interior of K. Show that
#(K ∩ Zn) ≤ 3n.
(Hint: For z = (z1, . . . , zn)ᵀ ∈ Zn ∩K consider the residue classes mod 3,i.e., the vectors z mod 3 = (z1 mod 3, . . . , zn mod 3)ᵀ).
Exercise 1.4 Let Λ ∈ L2. Show that there exists a basis b1, b2 of Λ suchthat
|〈b1, b2〉| ≤1
2|b1| |b2|.
Exercise 1.5 Let Γ(x) =∫∞0 tx−1e−tdt be the Γ-function. Show that for
1 ≤ p <∞
vol (Bpn) = 2n
Γ(1 + 1/p)n
Γ(1 + n/p).
Hint: For K ∈ Kno , dimK = n, and p ∈ [1,∞) it holds∫Rn
e−|x|pK dx = Γ(n/p+ 1) vol (K).