The Convex Hull of a Parametrized Curve Cynthia Vinzant Department - - PowerPoint PPT Presentation

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The Convex Hull of a Parametrized Curve Cynthia Vinzant Department - - PowerPoint PPT Presentation

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The Convex Hull of a Parametrized Curve Cynthia Vinzant Department of Mathematics University of California, Berkeley SIAM - Convex Algebraic Geometry July 15, 2010 Cynthia Vinzant The Convex Hull of a Parametrized Curve Faces and Vertices of

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The Convex Hull of a Parametrized Curve

Cynthia Vinzant

Department of Mathematics University of California, Berkeley

SIAM - Convex Algebraic Geometry

July 15, 2010

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Faces and Vertices of Convex Hulls

Let f(t) = (f1(t), . . . , fn(t)) where fj ∈ R[t] and let D ⊆ R. Our curve: C = {f(t) : t ∈ D}

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Faces and Vertices of Convex Hulls

Let f(t) = (f1(t), . . . , fn(t)) where fj ∈ R[t] and let D ⊆ R. Our curve: C = {f(t) : t ∈ D} Goal: Compute the set of (a1, . . . , ar) ∈ Dr where f(a1), . . . , f(ar) ∈ Rn are the vertices of a face of conv(C).

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Faces and Vertices of Convex Hulls

Let f(t) = (f1(t), . . . , fn(t)) where fj ∈ R[t] and let D ⊆ R. Our curve: C = {f(t) : t ∈ D} Goal: Compute the set of (a1, . . . , ar) ∈ Dr where f(a1), . . . , f(ar) ∈ Rn are the vertices of a face of conv(C). Example: C = {(t, 4t3 − 3t, 16t5 − 20t3 + 5t) : t ∈ [−1, 1]} (a1, a2) ↔ edge [f(a1), f(a2)] (a1, a2) ↔ facet [f(a1), f(a2), f(a3)]

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Many ways to represent the convex hull of a curve

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Many ways to represent the convex hull of a curve

Zariski Closure of ∂ conv(C) (Ranestad and Sturmfels, 2010):

  • works on general varieties, hard to compute
  • gives algebraic information

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Many ways to represent the convex hull of a curve

Zariski Closure of ∂ conv(C) (Ranestad and Sturmfels, 2010):

  • works on general varieties, hard to compute
  • gives algebraic information

Projection of a Spectrahedron (Henrion, 2010):

  • easy to compute, easy optimization
  • hard to recover algebra/faces of ∂ conv(C)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Many ways to represent the convex hull of a curve

Zariski Closure of ∂ conv(C) (Ranestad and Sturmfels, 2010):

  • works on general varieties, hard to compute
  • gives algebraic information

Projection of a Spectrahedron (Henrion, 2010):

  • easy to compute, easy optimization
  • hard to recover algebra/faces of ∂ conv(C)

Face-vertex set (this talk):

  • complete facial information of conv(C)
  • invariant under change of coordinates
  • medium-hard to compute

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Affine Functions ← → Polynomials

Affine functions on Rn ← → Polynomials in span{1, f1, . . . , fn} w0 + wtx ← → g(t) = w0 +

j wjfj(t)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Affine Functions ← → Polynomials

Affine functions on Rn ← → Polynomials in span{1, f1, . . . , fn} w0 + wtx ← → g(t) = w0 +

j wjfj(t)

Nonnegativity: The halfspace {wTx ≥ w0} contains the curve C = {f(t) : t ∈ D} if and only if the polynomial g(t) ≥ 0 on D. Equality: The intersection of the curve C and the plane {wTx = w0} is the set of points {f(a) : g(a) = 0}.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Affine Functions ← → Polynomials

Affine functions on Rn ← → Polynomials in span{1, f1, . . . , fn} w0 + wtx ← → g(t) = w0 +

j wjfj(t)

Nonnegativity: The halfspace {wTx ≥ w0} contains the curve C = {f(t) : t ∈ D} if and only if the polynomial g(t) ≥ 0 on D. Equality: The intersection of the curve C and the plane {wTx = w0} is the set of points {f(a) : g(a) = 0}. Faces: The points {f(a1), . . . , f(ar)} are the vertices of a face ⇔ there exists g ∈ span{1, f1, . . . , fn} with g ≥ 0 on D and {t ∈ D : g(t) = 0} = {a1, . . . , ar}

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Side note: Dual Bodies

The dual cone of conv(C) is {g ∈ span{1, fj} : g ≥ 0 on D}.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Side note: Dual Bodies

The dual cone of conv(C) is {g ∈ span{1, fj} : g ≥ 0 on D}.

Example 1: conv(1, t, t2) {b2 − 4ac ≤ 0, a ≥ 0}

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Side note: Dual Bodies

The dual cone of conv(C) is {g ∈ span{1, fj} : g ≥ 0 on D}.

Example 1: conv(1, t, t2) {b2 − 4ac ≤ 0, a ≥ 0}

Example 2: conv(t, 2t2 − 1, 4t3 − 3t)

{g ∈ R[t]≤3 : g ≥ 0 on [−1, 1]}

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Necessary Algebraic Conditions

  • Prop. For {a1, . . . , as} ⊂ int(D) and {as+1, . . . , ar} ∈ ∂D, TFAE:

Let Vr ⊂ Dr be the set of (a1, . . . , ar) satisfying these conditions.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Necessary Algebraic Conditions

  • Prop. For {a1, . . . , as} ⊂ int(D) and {as+1, . . . , ar} ∈ ∂D, TFAE:
  • 1. ∃ g ∈ span{1, f1 . . . , fn} with g(aj) = 0

and g′(aj) = 0 for j = 1, . . . , s Let Vr ⊂ Dr be the set of (a1, . . . , ar) satisfying these conditions.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Necessary Algebraic Conditions

  • Prop. For {a1, . . . , as} ⊂ int(D) and {as+1, . . . , ar} ∈ ∂D, TFAE:
  • 1. ∃ g ∈ span{1, f1 . . . , fn} with g(aj) = 0

and g′(aj) = 0 for j = 1, . . . , s

  • 2. {f(aj)} ∪ {f(aj) + f

′(aj) : aj ∈ int D}

lie in a common hyperplane Let Vr ⊂ Dr be the set of (a1, . . . , ar) satisfying these conditions.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Necessary Algebraic Conditions

  • Prop. For {a1, . . . , as} ⊂ int(D) and {as+1, . . . , ar} ∈ ∂D, TFAE:
  • 1. ∃ g ∈ span{1, f1 . . . , fn} with g(aj) = 0

and g′(aj) = 0 for j = 1, . . . , s

  • 2. {f(aj)} ∪ {f(aj) + f

′(aj) : aj ∈ int D}

lie in a common hyperplane

  • 3. rank
  • 1

. . . 1 1 . . . 1 f(a1) . . . f(ar) f

′(a1)

. . . f

′(as)

  • ≤ n

Let Vr ⊂ Dr be the set of (a1, . . . , ar) satisfying these conditions.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Exposing the real faces

“Discriminant”: S := π(Vr+1) ∪ sing(Vr) ∪ ... has codim-1 in Vr. To test which points in Vr\S are the vertices of a face on conv(C) it suffices to test one point in each connected component of Vr\S. Example: C = {(t, 4t3 − 3t, 16t5 − 20t3 + 5t) : t ∈ [−1, 1]}

V2 (potential edges) π(V3) (potential edges

  • f 2-faces)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Exposing the real faces

“Discriminant”: S := π(Vr+1) ∪ sing(Vr) ∪ ... has codim-1 in Vr. To test which points in Vr\S are the vertices of a face on conv(C) it suffices to test one point in each connected component of Vr\S. For fixed (aj)j ∈ Vr\S, this only involves testing whether a linear space in R[t] contains a polynomial g ≥ 0 on D. (an SDP!) Example: C = {(t, 4t3 − 3t, 16t5 − 20t3 + 5t) : t ∈ [−1, 1]}

V2 (potential edges) π(V3) (potential edges

  • f 2-faces)

V2\S (test points)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Exposing the real faces

“Discriminant”: S := π(Vr+1) ∪ sing(Vr) ∪ ... has codim-1 in Vr. To test which points in Vr\S are the vertices of a face on conv(C) it suffices to test one point in each connected component of Vr\S. For fixed (aj)j ∈ Vr\S, this only involves testing whether a linear space in R[t] contains a polynomial g ≥ 0 on D. (an SDP!) Example: C = {(t, 4t3 − 3t, 16t5 − 20t3 + 5t) : t ∈ [−1, 1]}

V2 (potential edges) π(V3) (potential edges

  • f 2-faces)

V2\S (test points)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Exposing the real faces

“Discriminant”: S := π(Vr+1) ∪ sing(Vr) ∪ ... has codim-1 in Vr. To test which points in Vr\S are the vertices of a face on conv(C) it suffices to test one point in each connected component of Vr\S. For fixed (aj)j ∈ Vr\S, this only involves testing whether a linear space in R[t] contains a polynomial g ≥ 0 on D. (an SDP!) Example: C = {(t, 4t3 − 3t, 16t5 − 20t3 + 5t) : t ∈ [−1, 1]}

V2 (potential edges) π(V3) (potential edges

  • f 2-faces)

→

V2\S (test points)

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Audience Challenge: Visualizing 4-dim’l convex bodies

Example: C = {(t, t3, t5, t7) : t ∈ [−1, 1]} ⊂ R4

  • •

edges of 4-faces l l l edges of 3-faces edges

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Stratification of the Grassmannian

For fixed D, the face-vertex sets of conv(C) depend only on span{1, f1(t), . . . , fn(t)} ⊂ R[t]. If deg(fj) ≤ d then span{f1, . . . , fn} is a point in Gr(n, R[t]≤d).

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Stratification of the Grassmannian

For fixed D, the face-vertex sets of conv(C) depend only on span{1, f1(t), . . . , fn(t)} ⊂ R[t]. If deg(fj) ≤ d then span{f1, . . . , fn} is a point in Gr(n, R[t]≤d). Example: C = {(f1(t), f2(t)) : t ∈ R} with deg(fj) ≤ 4.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Stratification of the Grassmannian

For fixed D, the face-vertex sets of conv(C) depend only on span{1, f1(t), . . . , fn(t)} ⊂ R[t]. If deg(fj) ≤ d then span{f1, . . . , fn} is a point in Gr(n, R[t]≤d). Example: C = {(f1(t), f2(t)) : t ∈ R} with deg(fj) ≤ 4. The set of planes in R[t]≤4 containing a polynomial (t − a)4 form a hypersurface H of in Gr(2, 4), given by

2p3

2,3−12p1,3p2,3p2,4+27p1,2p2 2,4+32p2 1,3p3,4−72p1,2p1,4p3,4−24p1,2p2,3p3,4

The number of edges of conv(C) is determined by what region

  • f Gr(2, 4)\H contains the plane span{f1, f2}.

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

◮ Given n, d, what are possible dimensions and degrees for Vr? ◮ What is the stratification of Gr(n, R[t]≤d)?

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

◮ Given n, d, what are possible dimensions and degrees for Vr? ◮ What is the stratification of Gr(n, R[t]≤d)? ◮ What can we easily recover about ∂ conv(C) ?

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

◮ Given n, d, what are possible dimensions and degrees for Vr? ◮ What is the stratification of Gr(n, R[t]≤d)? ◮ What can we easily recover about ∂ conv(C) ? ◮ What does conv(in f(t) : t ∈ R) tell us about

conv(f(t) : t ∈ R)?

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

◮ Given n, d, what are possible dimensions and degrees for Vr? ◮ What is the stratification of Gr(n, R[t]≤d)? ◮ What can we easily recover about ∂ conv(C) ? ◮ What does conv(in f(t) : t ∈ R) tell us about

conv(f(t) : t ∈ R)? Relation to SAGBI bases?

Cynthia Vinzant The Convex Hull of a Parametrized Curve

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Interesting Related Questions

◮ Given n, d, what are possible dimensions and degrees for Vr? ◮ What is the stratification of Gr(n, R[t]≤d)? ◮ What can we easily recover about ∂ conv(C) ? ◮ What does conv(in f(t) : t ∈ R) tell us about

conv(f(t) : t ∈ R)? Relation to SAGBI bases? Thanks!

Cynthia Vinzant The Convex Hull of a Parametrized Curve