On algebraic and geometric properties of spectral convex bodies - - PowerPoint PPT Presentation

On algebraic and geometric properties

• f spectral convex bodies

Raman Sanyal

Goethe-Universit¨ at Frankfurt

joint work with James Saunderson (Monash University) arXiv:2001.04361

1/ 17

Between you and I

How many convex bodies do you really know?

2/ 17

Between you and I

How many convex bodies do you really know?

[A convex body is a convex and compact set.]

2/ 17

Between you and I

How many convex bodies do you really know?

[A convex body is a convex and compact set.]

How many of them are not polytopes?

[A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.]

2/ 17

Between you and I

How many convex bodies do you really know?

[A convex body is a convex and compact set.]

How many of them are not polytopes?

[A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.]

Let me introduce you to

Spectral convex bodies

2/ 17

Between you and I

How many convex bodies do you really know?

[A convex body is a convex and compact set.]

How many of them are not polytopes?

[A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.]

Let me introduce you to

Spectral convex bodies

Focus today:

◮ Operations and metric properties

Minkowski sums, polarity, volume and Steiner polynomials

◮ Geometric and algebraic boundary

faces, algebraic degree, hyperbolicity

◮ Representations

spectrahedra and spectrahedral shadows

2/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) .

3/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) . A convex set K ⊆ Rd is symmetric if σK = K for all σ ∈ Sd.

3/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) . A convex set K ⊆ Rd is symmetric if σK = K for all σ ∈ Sd. Symmetric matrices S2Rd = {A ∈ Rd×d : At = A}.

3/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) . A convex set K ⊆ Rd is symmetric if σK = K for all σ ∈ Sd. Symmetric matrices S2Rd = {A ∈ Rd×d : At = A}. Spectrum λ(A) = (λ1, λ2, . . . , λd) ∈ Rd are the d real eigenvalues of A ∈ S2Rd.

3/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) . A convex set K ⊆ Rd is symmetric if σK = K for all σ ∈ Sd. Symmetric matrices S2Rd = {A ∈ Rd×d : At = A}. Spectrum λ(A) = (λ1, λ2, . . . , λd) ∈ Rd are the d real eigenvalues of A ∈ S2Rd. A spectral convex set is a set of the form Λ(K) := {A ∈ S2Rd : λ(A) ∈ K} , where K is a symmetric convex set.

3/ 17

Spectral convex sets

Sd group of permutations acting on Rd σ · (x1, . . . , xd) := (xσ(1), . . . , xσ(d)) . A convex set K ⊆ Rd is symmetric if σK = K for all σ ∈ Sd. Symmetric matrices S2Rd = {A ∈ Rd×d : At = A}. Spectrum λ(A) = (λ1, λ2, . . . , λd) ∈ Rd are the d real eigenvalues of A ∈ S2Rd. A spectral convex set is a set of the form Λ(K) := {A ∈ S2Rd : λ(A) ∈ K} , where K is a symmetric convex set.

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body.

3/ 17

Examples: Λ(K) := {A ∈ S2Rd : λ(A) ∈ K}

◮ Operator norm: K = {x : x∞ ≤ 1} = [−1, 1]d

Λ(K) = {A ∈ S2Rd : λmax(A) ≤ 1}

4/ 17

Examples: Λ(K) := {A ∈ S2Rd : λ(A) ∈ K}

◮ Operator norm: K = {x : x∞ ≤ 1} = [−1, 1]d

Λ(K) = {A ∈ S2Rd : λmax(A) ≤ 1}

◮ Nuclear norm: K = {x : x1 ≤ 1}

Λ(K) = {A ∈ S2Rd : |λ1(A)| + · · · + |λd(A)| ≤ 1}

4/ 17

Examples: Λ(K) := {A ∈ S2Rd : λ(A) ∈ K}

◮ Operator norm: K = {x : x∞ ≤ 1} = [−1, 1]d

Λ(K) = {A ∈ S2Rd : λmax(A) ≤ 1}

◮ Nuclear norm: K = {x : x1 ≤ 1}

Λ(K) = {A ∈ S2Rd : |λ1(A)| + · · · + |λd(A)| ≤ 1}

◮ Frobenius norm: K = B(Rd) = {x : x2 ≤ 1}

Λ(K) = {A ∈ S2Rd : λ1(A)2 + · · · + λd(A)2 ≤ 1} = B(S2Rd)

4/ 17

Examples: Λ(K) := {A ∈ S2Rd : λ(A) ∈ K}

◮ Operator norm: K = {x : x∞ ≤ 1} = [−1, 1]d

Λ(K) = {A ∈ S2Rd : λmax(A) ≤ 1}

◮ Nuclear norm: K = {x : x1 ≤ 1}

Λ(K) = {A ∈ S2Rd : |λ1(A)| + · · · + |λd(A)| ≤ 1}

◮ Frobenius norm: K = B(Rd) = {x : x2 ≤ 1}

Λ(K) = {A ∈ S2Rd : λ1(A)2 + · · · + λd(A)2 ≤ 1} = B(S2Rd)

◮ PSD cone: K = Rd ≥0

Λ(K) = {A ∈ S2Rd : A positive semidefinite}

4/ 17

Examples: Λ(K) := {A ∈ S2Rd : λ(A) ∈ K}

◮ Operator norm: K = {x : x∞ ≤ 1} = [−1, 1]d

Λ(K) = {A ∈ S2Rd : λmax(A) ≤ 1}

◮ Nuclear norm: K = {x : x1 ≤ 1}

Λ(K) = {A ∈ S2Rd : |λ1(A)| + · · · + |λd(A)| ≤ 1}

◮ Frobenius norm: K = B(Rd) = {x : x2 ≤ 1}

Λ(K) = {A ∈ S2Rd : λ1(A)2 + · · · + λd(A)2 ≤ 1} = B(S2Rd)

◮ PSD cone: K = Rd ≥0

Λ(K) = {A ∈ S2Rd : A positive semidefinite}

◮ Schur-Horn orbitopes [S-Sottile-Sturmfels’11]: K = conv(Sd · p)

Λ(K) = {A ∈ S2Rd : λ(A) majorized by p}

4/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body.

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

Proof of Proposition.

◮ Show A ∈ conv(Λ(K)), then A ∈ Λ(K).

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

Proof of Proposition.

◮ Show A ∈ conv(Λ(K)), then A ∈ Λ(K). ◮ O(d)-invariance: gΛ(K)g t = Λ(K) for all g ∈ O(d).

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

Proof of Proposition.

◮ Show A ∈ conv(Λ(K)), then A ∈ Λ(K). ◮ O(d)-invariance: gΛ(K)g t = Λ(K) for all g ∈ O(d). ◮ Wlog A = δ(p) diagonal.

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

Proof of Proposition.

◮ Show A ∈ conv(Λ(K)), then A ∈ Λ(K). ◮ O(d)-invariance: gΛ(K)g t = Λ(K) for all g ∈ O(d). ◮ Wlog A = δ(p) diagonal. ◮ A = i µiAi for Ai ∈ Λ(K) implies p = D(A) = i µiD(Ai) ∈ K.

5/ 17

Spectral convex sets

Proposition

If K is a symmetric convex set/body, then Λ(K) is a convex set/body. D : S2Rd → Rd diagonal projection D(A) = (A11, A22, . . . , Add) δ : Rd → S2Rd diagonal embedding

Lemma

Let K be a symmetric convex set. Then D(Λ(K)) = K = D(Λ(K) ∩ δ(Rd)) .

[Needs Schur’s insight: D(A) is majorized by λ(A).]

Proof of Proposition.

◮ Show A ∈ conv(Λ(K)), then A ∈ Λ(K). ◮ O(d)-invariance: gΛ(K)g t = Λ(K) for all g ∈ O(d). ◮ Wlog A = δ(p) diagonal. ◮ A = i µiAi for Ai ∈ Λ(K) implies p = D(A) = i µiD(Ai) ∈ K. ◮ Again by Lemma: A = δ(p) ∈ Λ(K).

5/ 17

Rich class of convex sets

Let K, L ⊂ Rd be symmetric closed convex sets.

◮ Intersections

Λ(K) ∩ Λ(L) = Λ(K ∩ L) .

6/ 17

Rich class of convex sets

Let K, L ⊂ Rd be symmetric closed convex sets.

◮ Intersections

Λ(K) ∩ Λ(L) = Λ(K ∩ L) .

◮ Minkowski sums: K + L := {p + q : p ∈ K, q ∈ L}

Λ(K) + Λ(L) = Λ(K + L) .

6/ 17

Rich class of convex sets

Let K, L ⊂ Rd be symmetric closed convex sets.

◮ Intersections

Λ(K) ∩ Λ(L) = Λ(K ∩ L) .

◮ Minkowski sums: K + L := {p + q : p ∈ K, q ∈ L}

Λ(K) + Λ(L) = Λ(K + L) .

◮ Convex hulls: K ∨ L := conv(K ∪ L)

Λ(K) ∨ Λ(L) = Λ(K ∨ L) . [Hint: K ∨ L =

0≤µ≤1(1 − µ)K + µL. ]

6/ 17

Rich class of convex sets

Let K, L ⊂ Rd be symmetric closed convex sets.

◮ Intersections

Λ(K) ∩ Λ(L) = Λ(K ∩ L) .

◮ Minkowski sums: K + L := {p + q : p ∈ K, q ∈ L}

Λ(K) + Λ(L) = Λ(K + L) .

◮ Convex hulls: K ∨ L := conv(K ∪ L)

Λ(K) ∨ Λ(L) = Λ(K ∨ L) . [Hint: K ∨ L =

0≤µ≤1(1 − µ)K + µL. ] ◮ Polarity: K ◦ = {c ∈ Rd : c, x ≤ 1 for all x ∈ K}

Λ(K)◦ = Λ(K ◦) .

6/ 17

Support functions

For closed convex set K ⊂ Rd, the support function is hK : Rd → R hK(c) := max{c, x : x ∈ K}

7/ 17

Support functions

For closed convex set K ⊂ Rd, the support function is hK : Rd → R hK(c) := max{c, x : x ∈ K} hK encodes K: K = {x : c, x ≤ hK(c) for all c} .

7/ 17

Support functions

For closed convex set K ⊂ Rd, the support function is hK : Rd → R hK(c) := max{c, x : x ∈ K} hK encodes K: K = {x : c, x ≤ hK(c) for all c} . Frobenius inner product on S2Rd: A, B = tr(AB).

Lemma

If K is closed symmetric convex set and B ∈ S2Rd, then hΛ(K)(B) = hK(λ(B)) . hΛ(K) is a spectral convex function in the sense of [Lewis’96].

7/ 17

Support functions

For closed convex set K ⊂ Rd, the support function is hK : Rd → R hK(c) := max{c, x : x ∈ K} hK encodes K: K = {x : c, x ≤ hK(c) for all c} . Frobenius inner product on S2Rd: A, B = tr(AB).

Lemma

If K is closed symmetric convex set and B ∈ S2Rd, then hΛ(K)(B) = hK(λ(B)) . hΛ(K) is a spectral convex function in the sense of [Lewis’96].

◮ Minkowski sum: Λ(K) + Λ(L) = Λ(K + L)

hK+L = hK + hL

◮ Polarity: Λ(K)◦ = Λ(K ◦)

K ◦ = {c : hK(c) ≤ 1}

7/ 17

Computation of convex invariants

K ⊂ Rd convex body, Bd = B(Rd) unit ball Steiner polynomial vol(K + t · Bd) =

d

• i=0

d i

• Wd−i(K)ti

8/ 17

Computation of convex invariants

K ⊂ Rd convex body, Bd = B(Rd) unit ball Steiner polynomial vol(K + t · Bd) =

d

• i=0

d i

• Wd−i(K)ti

Quermaßintegrals Wi(K) ∼ expected volume of projection onto (d − i)-flat Wd volume, Wd−1 surface area, W1 mean width, W0 Euler characteristic Important invariants, hard to compute

8/ 17

Computation of convex invariants

K ⊂ Rd convex body, Bd = B(Rd) unit ball Steiner polynomial vol(K + t · Bd) =

d

• i=0

d i

• Wd−i(K)ti

Quermaßintegrals Wi(K) ∼ expected volume of projection onto (d − i)-flat Wd volume, Wd−1 surface area, W1 mean width, W0 Euler characteristic Important invariants, hard to compute

Theorem

vol(Λ(K) + t · B(S2Rd)) = 2

d(d+3) 2

d

• r=1

π

r 2

Γ( r

2)

• K+tBd
• i<j

|pi − pj|dp If K symmetric polytope, then integral effectively computable.

8/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K).

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K). Algebraic boundary ∂algK is Zariski closure of ∂K, given by fK ∈ R[x1, . . . , xd].

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K). Algebraic boundary ∂algK is Zariski closure of ∂K, given by fK ∈ R[x1, . . . , xd]. If K symmetric, then fK symmetric fK(σ · x) = fK(x) for all σ ∈ Sd. fK = FK(e1, . . . , ed), where ei are elementary symmetric polynomials.

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K). Algebraic boundary ∂algK is Zariski closure of ∂K, given by fK ∈ R[x1, . . . , xd]. If K symmetric, then fK symmetric fK(σ · x) = fK(x) for all σ ∈ Sd. fK = FK(e1, . . . , ed), where ei are elementary symmetric polynomials. det(A + tI) = td + η1(A)td−1 + · · · + ηd(A) ηi(A) fundamental invariants of O(d)-action on S2Rd (sums of principal minors)

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K). Algebraic boundary ∂algK is Zariski closure of ∂K, given by fK ∈ R[x1, . . . , xd]. If K symmetric, then fK symmetric fK(σ · x) = fK(x) for all σ ∈ Sd. fK = FK(e1, . . . , ed), where ei are elementary symmetric polynomials. det(A + tI) = td + η1(A)td−1 + · · · + ηd(A) ηi(A) fundamental invariants of O(d)-action on S2Rd (sums of principal minors)

Proposition

fΛ(K)(A) = FK(η1(A), . . . , ηd(A))

9/ 17

Faces and boundaries

(Exposed) Face of full-dimensional convex body K ⊂ Rd K c := {x ∈ K : c, x = hK(c)} for some c ∈ Rd

Proposition

Sd-orbits of faces of K

1-to-1

← → O(d)-orbits of faces of Λ(K). Algebraic boundary ∂algK is Zariski closure of ∂K, given by fK ∈ R[x1, . . . , xd]. If K symmetric, then fK symmetric fK(σ · x) = fK(x) for all σ ∈ Sd. fK = FK(e1, . . . , ed), where ei are elementary symmetric polynomials. det(A + tI) = td + η1(A)td−1 + · · · + ηd(A) ηi(A) fundamental invariants of O(d)-action on S2Rd (sums of principal minors)

Proposition

fΛ(K)(A) = FK(η1(A), . . . , ηd(A))

Corollary

∂algΛ(K) and ∂algK have same degree. If K hyperbolicity cone, then Λ(K) is hyperbolicity cone [Bauschke et al.’01]

9/ 17

Spectrahedra

Spectrahedron S = {x ∈ Rd : B(x) := B0 + x1B1 + · · · + xdBd 0} , where B0, . . . , Bd ∈ S2Rd and 0 means positive semidefinite.

10/ 17

Spectrahedra

Spectrahedron S = {x ∈ Rd : B(x) := B0 + x1B1 + · · · + xdBd 0} , where B0, . . . , Bd ∈ S2Rd and 0 means positive semidefinite. Semidefinite programming max c, x

• subj. to B(x) 0

generalizes linear programming, fast algorithms

10/ 17

Spectrahedra

Spectrahedron S = {x ∈ Rd : B(x) := B0 + x1B1 + · · · + xdBd 0} , where B0, . . . , Bd ∈ S2Rd and 0 means positive semidefinite. Semidefinite programming max c, x

• subj. to B(x) 0

generalizes linear programming, fast algorithms generalize polyhedra, many favorable properties, many important examples

10/ 17

Spectrahedra

Spectrahedron S = {x ∈ Rd : B(x) := B0 + x1B1 + · · · + xdBd 0} , where B0, . . . , Bd ∈ S2Rd and 0 means positive semidefinite. Semidefinite programming max c, x

• subj. to B(x) 0

generalizes linear programming, fast algorithms generalize polyhedra, many favorable properties, many important examples For example operator/nuclear/Frobenius unit balls are spectrahedra

10/ 17

Spectrahedra

Spectrahedron S = {x ∈ Rd : B(x) := B0 + x1B1 + · · · + xdBd 0} , where B0, . . . , Bd ∈ S2Rd and 0 means positive semidefinite. Semidefinite programming max c, x

• subj. to B(x) 0

generalizes linear programming, fast algorithms generalize polyhedra, many favorable properties, many important examples For example operator/nuclear/Frobenius unit balls are spectrahedra

Theorem

If K is a polyhedron, then Λ(K) is a spectrahedron.

10/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

11/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

Proposition

x ∈ Π(p) if and only if x majorized by p. That is,

i xi = i pi |J|

• i=1

pi ≥

• i∈J

xi for all J ⊆ {1, . . . , d}.

11/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

Proposition

x ∈ Π(p) if and only if x majorized by p. That is,

i xi = i pi |J|

• i=1

pi ≥

• i∈J

xi for all J ⊆ {1, . . . , d}. k-th linearized Schur functor: linear map Lk : S2Rd → S2(k Rd)

11/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

Proposition

x ∈ Π(p) if and only if x majorized by p. That is,

i xi = i pi |J|

• i=1

pi ≥

• i∈J

xi for all J ⊆ {1, . . . , d}. k-th linearized Schur functor: linear map Lk : S2Rd → S2(k Rd) d

k

• eigenvalues of Lk(A) are

i∈J λi(A) for |J| = k.

11/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

Proposition

x ∈ Π(p) if and only if x majorized by p. That is,

i xi = i pi |J|

• i=1

pi ≥

• i∈J

xi for all J ⊆ {1, . . . , d}. k-th linearized Schur functor: linear map Lk : S2Rd → S2(k Rd) d

k

• eigenvalues of Lk(A) are

i∈J λi(A) for |J| = k.

Theorem (S-Sottile-Sturmfels’11)

Let A ∈ S2Rd. Then A ∈ Λ(Π(p)) if and only if tr(A) =

i pi and

(p1 + · · · + pk) · Id − Lk(A) 0 for all 0 < k < d.

11/ 17

Schur-Horn orbitopes are spectrahedra

Let p = (p1 ≥ p2 ≥ · · · ≥ pd) Permutahedron Π(p) = conv

• (pσ(1), . . . , pσ(d)) : σ ∈ Sd
• .

Proposition

x ∈ Π(p) if and only if x majorized by p. That is,

i xi = i pi |J|

• i=1

pi ≥

• i∈J

xi for all J ⊆ {1, . . . , d}. k-th linearized Schur functor: linear map Lk : S2Rd → S2(k Rd) d

k

• eigenvalues of Lk(A) are

i∈J λi(A) for |J| = k.

Theorem (S-Sottile-Sturmfels’11)

Let A ∈ S2Rd. Then A ∈ Λ(Π(p)) if and only if tr(A) =

i pi and

(p1 + · · · + pk) · Id − Lk(A) 0 for all 0 < k < d. Schur-Horn orbitope is a spectrahedron of order 2d − 2. For p generic, algebraic degree of Π(p) is 2d − 2.

11/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} .

12/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,bi

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron.

12/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,bi

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron.

Question

For a ∈ Rd, is there a linear map La : S2Rd → S2Rd! such that the d! eigenvalues of La(A) are σ · a, λ(A) for σ ∈ Sd?

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Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,bi

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron.

Question

For a ∈ Rd, is there a linear map La : S2Rd → S2Rd! such that the d! eigenvalues of La(A) are σ · a, λ(A) for σ ∈ Sd? If yes, then Λ(Pa,b) = {A ∈ S2Rd : b · Id − La(A) 0} .

12/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,bi

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron.

Question

For a ∈ Rd, is there a linear map La : S2Rd → S2Rd! such that the d! eigenvalues of La(A) are σ · a, λ(A) for σ ∈ Sd? If yes, then Λ(Pa,b) = {A ∈ S2Rd : b · Id − La(A) 0} . Yes, for d = 2: A → La(A) = a1A + a2Aadj

12/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,b

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron. Assume a = (a1 ≥ a2 ≥ · · · ≥ ad)

13/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,b

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron. Assume a = (a1 ≥ a2 ≥ · · · ≥ ad) numerical chain I = (I1, I2, . . . , Id) where Ik is k-subset of {1, . . . , d}

13/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,b

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron. Assume a = (a1 ≥ a2 ≥ · · · ≥ ad) numerical chain I = (I1, I2, . . . , Id) where Ik is k-subset of {1, . . . , d} Define aI := (a1 − a2)1I1 + (a2 − a3)1I1 + · · · + ad1Id , where 1Ij ∈ {0, 1}d are characteristic vectors.

13/ 17

Spectral polyhedra

a ∈ Rd, b ∈ R define Pa,b = {x ∈ Rd : σ · a, x ≤ b for all σ ∈ Sd} . Symmetric polyhedra are of the form P =

i Pai,b

Since Λ(K ∩ L) = Λ(K) ∩ Λ(L), suffices to show that Λ(Pa,b) is spectrahedron. Assume a = (a1 ≥ a2 ≥ · · · ≥ ad) numerical chain I = (I1, I2, . . . , Id) where Ik is k-subset of {1, . . . , d} Define aI := (a1 − a2)1I1 + (a2 − a3)1I1 + · · · + ad1Id , where 1Ij ∈ {0, 1}d are characteristic vectors. If I is a chain, that is, I1 ⊂ I2 ⊂ · · · ⊂ Id, then aI = σ · a for some σ ∈ Sd.

Proposition

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} .

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Spectral polyhedra

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} .

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Spectral polyhedra

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} . Recall that for A, B ∈ S2Rd, A ⊗ B is symmetric d2 × d2-matrix with eigenvalues λi(A) · λj(B) for 1 ≤ i, j ≤ d

14/ 17

Spectral polyhedra

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} . Recall that for A, B ∈ S2Rd, A ⊗ B is symmetric d2 × d2-matrix with eigenvalues λi(A) · λj(B) for 1 ≤ i, j ≤ d Define La : 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd → 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd by

14/ 17

Spectral polyhedra

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} . Recall that for A, B ∈ S2Rd, A ⊗ B is symmetric d2 × d2-matrix with eigenvalues λi(A) · λj(B) for 1 ≤ i, j ≤ d Define La : 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd → 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd by

• La(A) :=

d

• j=1

(aj − aj+1)Id1 Rd ⊗ · · · ⊗ Idj−1 Rd ⊗ Lj(A) ⊗ · · · ⊗ Idd Rd Eigenvalues of La(A) are aI, λ(A) for all I numerical chains.

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Spectral polyhedra

Pa,b = {x ∈ Rd : aI, x ≤ b for I numerical chain} . Recall that for A, B ∈ S2Rd, A ⊗ B is symmetric d2 × d2-matrix with eigenvalues λi(A) · λj(B) for 1 ≤ i, j ≤ d Define La : 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd → 1 Rd ⊗ 2 Rd ⊗ · · · ⊗ d Rd by

• La(A) :=

d

• j=1

(aj − aj+1)Id1 Rd ⊗ · · · ⊗ Idj−1 Rd ⊗ Lj(A) ⊗ · · · ⊗ Idd Rd Eigenvalues of La(A) are aI, λ(A) for all I numerical chains.

Theorem

If P = Pa1,b1 ∩ · · · ∩ PaM,bM is a symmetric polyhedron, then Λ(P) = {A ∈ S2Rd : bi · Id − Lai(A) 0 for i = 1, . . . , M} .

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Spectrahedra shadows

Λ(P) = {A ∈ S2Rd : bi · Id − Lai(A) 0 for i = 1, . . . , M} is a spectrahedron of order M · 2d2.

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Spectrahedra shadows

Λ(P) = {A ∈ S2Rd : bi · Id − Lai(A) 0 for i = 1, . . . , M} is a spectrahedron of order M · 2d2. In contrast: Λ(P) has algebraic degree M · d!.

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Spectrahedra shadows

Λ(P) = {A ∈ S2Rd : bi · Id − Lai(A) 0 for i = 1, . . . , M} is a spectrahedron of order M · 2d2. In contrast: Λ(P) has algebraic degree M · d!. Spectrahedral shadow is a projection of spectrahedron S = {x ∈ Rd : ∃y ∈ Re such that B(x) + B′(y) 0} . In contrast to polyhedra, spectrahedral shadows are in general not spectrahedra.

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Spectrahedra shadows

Λ(P) = {A ∈ S2Rd : bi · Id − Lai(A) 0 for i = 1, . . . , M} is a spectrahedron of order M · 2d2. In contrast: Λ(P) has algebraic degree M · d!. Spectrahedral shadow is a projection of spectrahedron S = {x ∈ Rd : ∃y ∈ Re such that B(x) + B′(y) 0} . In contrast to polyhedra, spectrahedral shadows are in general not spectrahedra. Using a result of Ben-Tal and Nemirovski, we can show

Theorem

If K ⊂ Rd is a symmetric spectrahedral shadow of order r, then Λ(P) is the projection of a spectrahedron of order r + 2d2 − 2d − 2. Similarly, if K is a symmetric polyhedron with r orbits of facets.

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Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0

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Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm]

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Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm] Important in convex geometry, geometric combinatorics, spline theory, ... The standard permutahedron for p = (1, 2, . . . , d) Π(p) = conv(Sd · p) = p +

• 1≤i<j≤d

[−(ei − ej), ei − ej] Nice formulas for volumes and Steiner polynomials, ...

16/ 17

Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm] Important in convex geometry, geometric combinatorics, spline theory, ... The standard permutahedron for p = (1, 2, . . . , d) Π(p) = conv(Sd · p) = p +

• 1≤i<j≤d

[−(ei − ej), ei − ej] Nice formulas for volumes and Steiner polynomials, ... Encode arrangement of hyperplanes Hi = z⊥

i

for i = 1, . . . , m

16/ 17

Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm] Important in convex geometry, geometric combinatorics, spline theory, ... The standard permutahedron for p = (1, 2, . . . , d) Π(p) = conv(Sd · p) = p +

• 1≤i<j≤d

[−(ei − ej), ei − ej] Nice formulas for volumes and Steiner polynomials, ... Encode arrangement of hyperplanes Hi = z⊥

i

for i = 1, . . . , m If Z is a symmetric zonotope, then Λ(Z) is a spectral zonotope.

16/ 17

Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm] Important in convex geometry, geometric combinatorics, spline theory, ... The standard permutahedron for p = (1, 2, . . . , d) Π(p) = conv(Sd · p) = p +

• 1≤i<j≤d

[−(ei − ej), ei − ej] Nice formulas for volumes and Steiner polynomials, ... Encode arrangement of hyperplanes Hi = z⊥

i

for i = 1, . . . , m If Z is a symmetric zonotope, then Λ(Z) is a spectral zonotope. Encode spectral arrangements {A ∈ S2Rd : σ · zi, λ(A) = 0 for some σ ∈ Sd}

16/ 17

Spectral zonotopes and spectral arrangements?

Line segment [−z, z] for z ∈ Rd \ 0 Zonotope Z = [−z1, z1] + · · · + [−zm, zm] Important in convex geometry, geometric combinatorics, spline theory, ... The standard permutahedron for p = (1, 2, . . . , d) Π(p) = conv(Sd · p) = p +

• 1≤i<j≤d

[−(ei − ej), ei − ej] Nice formulas for volumes and Steiner polynomials, ... Encode arrangement of hyperplanes Hi = z⊥

i

for i = 1, . . . , m If Z is a symmetric zonotope, then Λ(Z) is a spectral zonotope. Encode spectral arrangements {A ∈ S2Rd : σ · zi, λ(A) = 0 for some σ ∈ Sd}

Question

Is there a nice theory of spectral zonotopes and spectral arrangements?

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Spectral convex bodies

A spectral convex set is a set of the form Λ(K) := {A ∈ S2Rd : λ(A) ∈ K} , where K is a symmetric convex set.

Proposition

Λ(K) is a convex set/body.

◮ Rich class of convex sets Closed under intersection Minkowski sum, convex

hull, and polarity.

◮ Geometric and algebraic structure intimately related to that of K

Support functions, orbits of faces, algebraic boundary, hyperbolicity

◮ Representations as (projections of) spectrahedra spectrahedra when K

polyhedron; spectrahedral shadow when K spectrahedral shadow

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