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

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 Convex Hulls Let f ( t ) = ( f 1 ( t ) , . . . , f n ( t )) where f j ∈ R [ t ] and let D ⊆ R . Our curve: C = { f ( t ) : t ∈ D} Cynthia Vinzant The Convex Hull of a Parametrized Curve

Faces and Vertices of Convex Hulls Let f ( t ) = ( f 1 ( t ) , . . . , f n ( t )) where f j ∈ R [ t ] and let D ⊆ R . Our curve: C = { f ( t ) : t ∈ D} Goal: Compute the set of ( a 1 , . . . , a r ) ∈ D r where f ( a 1 ) , . . . , f ( a r ) ∈ R n are the vertices of a face of conv( C ). Cynthia Vinzant The Convex Hull of a Parametrized Curve

Faces and Vertices of Convex Hulls Let f ( t ) = ( f 1 ( t ) , . . . , f n ( t )) where f j ∈ R [ t ] and let D ⊆ R . Our curve: C = { f ( t ) : t ∈ D} Goal: Compute the set of ( a 1 , . . . , a r ) ∈ D r where f ( a 1 ) , . . . , f ( a r ) ∈ R n are the vertices of a face of conv( C ). Example: C = { ( t , 4 t 3 − 3 t , 16 t 5 − 20 t 3 + 5 t ) : t ∈ [ − 1 , 1] } ( a 1 , a 2 ) ↔ edge [ f ( a 1 ) , f ( a 2 )] ( a 1 , a 2 ) ↔ facet [ f ( a 1 ) , f ( a 2 ) , f ( a 3 )] Cynthia Vinzant The Convex Hull of a Parametrized Curve

Many ways to represent the convex hull of a curve Cynthia Vinzant The Convex Hull of a Parametrized Curve

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

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

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

Affine Functions ← → Polynomials Affine functions on R n ← → Polynomials in span { 1 , f 1 , . . . , f n } w 0 + w t x ← → g ( t ) = w 0 + � j w j f j ( t ) Cynthia Vinzant The Convex Hull of a Parametrized Curve

Affine Functions ← → Polynomials Affine functions on R n ← → Polynomials in span { 1 , f 1 , . . . , f n } w 0 + w t x ← → g ( t ) = w 0 + � j w j f j ( t ) Nonnegativity: The halfspace { w T x ≥ w 0 } 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 { w T x = w 0 } is the set of points { f ( a ) : g ( a ) = 0 } . Cynthia Vinzant The Convex Hull of a Parametrized Curve

Affine Functions ← → Polynomials Affine functions on R n ← → Polynomials in span { 1 , f 1 , . . . , f n } w 0 + w t x ← → g ( t ) = w 0 + � j w j f j ( t ) Nonnegativity: The halfspace { w T x ≥ w 0 } 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 { w T x = w 0 } is the set of points { f ( a ) : g ( a ) = 0 } . Faces: The points { f ( a 1 ) , . . . , f ( a r ) } are the vertices of a face ⇔ there exists g ∈ span { 1 , f 1 , . . . , f n } with g ≥ 0 on D and { t ∈ D : g ( t ) = 0 } = { a 1 , . . . , a r } Cynthia Vinzant The Convex Hull of a Parametrized Curve

Side note: Dual Bodies The dual cone of conv( C ) is { g ∈ span { 1 , f j } : g ≥ 0 on D} . Cynthia Vinzant The Convex Hull of a Parametrized Curve

Side note: Dual Bodies The dual cone of conv( C ) is { g ∈ span { 1 , f j } : g ≥ 0 on D} . { b 2 − 4 ac ≤ 0 , a ≥ 0 } conv (1 , t , t 2 ) Example 1: Cynthia Vinzant The Convex Hull of a Parametrized Curve

Side note: Dual Bodies The dual cone of conv( C ) is { g ∈ span { 1 , f j } : g ≥ 0 on D} . { b 2 − 4 ac ≤ 0 , a ≥ 0 } conv (1 , t , t 2 ) Example 1: Example 2: conv ( t , 2 t 2 − 1 , 4 t 3 − 3 t ) { g ∈ R [ t ] ≤ 3 : g ≥ 0 on [ − 1 , 1] } Cynthia Vinzant The Convex Hull of a Parametrized Curve

Necessary Algebraic Conditions Prop. For { a 1 , . . . , a s } ⊂ int( D ) and { a s +1 , . . . , a r } ∈ ∂ D , TFAE: Let V r ⊂ D r be the set of ( a 1 , . . . , a r ) satisfying these conditions. Cynthia Vinzant The Convex Hull of a Parametrized Curve

Necessary Algebraic Conditions Prop. For { a 1 , . . . , a s } ⊂ int( D ) and { a s +1 , . . . , a r } ∈ ∂ D , TFAE: 1. ∃ g ∈ span { 1 , f 1 . . . , f n } with g ( a j ) = 0 and g ′ ( a j ) = 0 for j = 1 , . . . , s Let V r ⊂ D r be the set of ( a 1 , . . . , a r ) satisfying these conditions. Cynthia Vinzant The Convex Hull of a Parametrized Curve

Necessary Algebraic Conditions Prop. For { a 1 , . . . , a s } ⊂ int( D ) and { a s +1 , . . . , a r } ∈ ∂ D , TFAE: 1. ∃ g ∈ span { 1 , f 1 . . . , f n } with g ( a j ) = 0 and g ′ ( a j ) = 0 for j = 1 , . . . , s ′ ( a j ) : a j ∈ int D} 2. { f ( a j ) } ∪ { f ( a j ) + f lie in a common hyperplane Let V r ⊂ D r be the set of ( a 1 , . . . , a r ) satisfying these conditions. Cynthia Vinzant The Convex Hull of a Parametrized Curve

Necessary Algebraic Conditions Prop. For { a 1 , . . . , a s } ⊂ int( D ) and { a s +1 , . . . , a r } ∈ ∂ D , TFAE: 1. ∃ g ∈ span { 1 , f 1 . . . , f n } with g ( a j ) = 0 and g ′ ( a j ) = 0 for j = 1 , . . . , s ′ ( a j ) : a j ∈ int D} 2. { f ( a j ) } ∪ { f ( a j ) + f lie in a common hyperplane �� 1 . . . 1 1 . . . 1 �� 3. rank ≤ n ′ ( a 1 ) ′ ( a s ) f ( a 1 ) . . . f ( a r ) f . . . f Let V r ⊂ D r be the set of ( a 1 , . . . , a r ) satisfying these conditions. Cynthia Vinzant The Convex Hull of a Parametrized Curve

Exposing the real faces “Discriminant”: S := π ( V r +1 ) ∪ sing( V r ) ∪ ... has codim-1 in V r . To test which points in V r \S are the vertices of a face on conv( C ) it suffices to test one point in each connected component of V r \S . Example: C = { ( t , 4 t 3 − 3 t , 16 t 5 − 20 t 3 + 5 t ) : t ∈ [ − 1 , 1] } V 2 (potential edges) π ( V 3 ) (potential edges of 2-faces) Cynthia Vinzant The Convex Hull of a Parametrized Curve

Exposing the real faces “Discriminant”: S := π ( V r +1 ) ∪ sing( V r ) ∪ ... has codim-1 in V r . To test which points in V r \S are the vertices of a face on conv( C ) it suffices to test one point in each connected component of V r \S . For fixed ( a j ) j ∈ V r \S , this only involves testing whether a linear space in R [ t ] contains a polynomial g ≥ 0 on D . (an SDP!) Example: C = { ( t , 4 t 3 − 3 t , 16 t 5 − 20 t 3 + 5 t ) : t ∈ [ − 1 , 1] } V 2 (potential edges) π ( V 3 ) (potential edges of 2-faces) V 2 \S (test points) Cynthia Vinzant The Convex Hull of a Parametrized Curve

Exposing the real faces “Discriminant”: S := π ( V r +1 ) ∪ sing( V r ) ∪ ... has codim-1 in V r . To test which points in V r \S are the vertices of a face on conv( C ) it suffices to test one point in each connected component of V r \S . For fixed ( a j ) j ∈ V r \S , this only involves testing whether a linear space in R [ t ] contains a polynomial g ≥ 0 on D . (an SDP!) Example: C = { ( t , 4 t 3 − 3 t , 16 t 5 − 20 t 3 + 5 t ) : t ∈ [ − 1 , 1] } V 2 (potential edges) π ( V 3 ) (potential edges of 2-faces) V 2 \S (test points) Cynthia Vinzant The Convex Hull of a Parametrized Curve

Exposing the real faces “Discriminant”: S := π ( V r +1 ) ∪ sing( V r ) ∪ ... has codim-1 in V r . To test which points in V r \S are the vertices of a face on conv( C ) it suffices to test one point in each connected component of V r \S . For fixed ( a j ) j ∈ V r \S , this only involves testing whether a linear space in R [ t ] contains a polynomial g ≥ 0 on D . (an SDP!) Example: C = { ( t , 4 t 3 − 3 t , 16 t 5 − 20 t 3 + 5 t ) : t ∈ [ − 1 , 1] } V 2 (potential edges) π ( V 3 ) (potential edges → of 2-faces) V 2 \S (test points) Cynthia Vinzant The Convex Hull of a Parametrized Curve

Audience Challenge: Visualizing 4-dim’l convex bodies Example: C = { ( t , t 3 , t 5 , t 7 ) : t ∈ [ − 1 , 1] } ⊂ R 4 • • edges of 4-faces l l l edges of 3-faces edges Cynthia Vinzant The Convex Hull of a Parametrized Curve

Stratification of the Grassmannian For fixed D , the face-vertex sets of conv( C ) depend only on span { 1 , f 1 ( t ) , . . . , f n ( t ) } ⊂ R [ t ]. If deg( f j ) ≤ d then span { f 1 , . . . , f n } is a point in Gr( n , R [ t ] ≤ d ). Cynthia Vinzant The Convex Hull of a Parametrized Curve