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 Berlin Martin Henk Discrete Geometry III SS 2015 Exercise sheet 3 Discussion: Wednesday, 06.05.2015. Exercise 3.1 Let Λ ∈L n and let L n-1 ∈L(n -1, Λ). Show that there exists a primitive vector b ? Λ * with i) L n-1 = {x R n : hb ? , xi =0}. ii) det(Λ L n-1 ) = det Λ ·|b * |. Exercise 3.2 Let Λ 0 Λ ∈L n be a sublattice, let a Λ 0 be primitive w.r.t. Λ 0 , and let α N such that 1 α a is primitive w.r.t. Λ. Show that α is a divisor of the index [Λ : Λ 0 ]. Exercise 3.3 Let k N 1 , K ∈K n 0 and let Λ ∈L n with vol (K) k 2 n det Λ. Then #(K Λ) 2k +1. Exercise 3.4 Let P R n be a lattice polytope having k interior lattice points of the integral lattice Z n . Show that vol (P ) nk +1 n! . Is this inequality best possible?

Transcript of Exercise sheet 3 - TU Berlin€¦ · Exercise 3.2 Let 0 ˆ 2Ln be a sublattice, let a 2 0 be...

Page 1: Exercise sheet 3 - TU Berlin€¦ · Exercise 3.2 Let 0 ˆ 2Ln be a sublattice, let a 2 0 be primitive w.r.t. 0, and let 2N such that 1 a is primitive w.r.t. . Show that is a divisor

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?

Page 2: Exercise sheet 3 - TU Berlin€¦ · Exercise 3.2 Let 0 ˆ 2Ln be a sublattice, let a 2 0 be primitive w.r.t. 0, and let 2N such that 1 a is primitive w.r.t. . Show that is a divisor

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?

Page 3: Exercise sheet 3 - TU Berlin€¦ · Exercise 3.2 Let 0 ˆ 2Ln be a sublattice, let a 2 0 be primitive w.r.t. 0, and let 2N such that 1 a is primitive w.r.t. . Show that is a divisor

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).