Seminar of computational geometry

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Seminar of computational geometry

• Lecture 1 Convexity

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Example of coordinate-dependence
Point p
Point q

• What is the sum of these two positions ?

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If you assume coordinates,
p (x1, y1)
q (x2, y2)

• The sum is (x1x2, y1y2)
• Is it correct ?
• Is it geometrically meaningful ?

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If you assume coordinates,
p (x1, y1)
(x1x2, y1y2)
q (x2, y2)
Origin

• Vector sum
• (x1, y1) and (x2, y2) are considered as vectors
  from the origin to p and q, respectively.

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If you select a different origin,
p (x1, y1)
(x1x2, y1y2)
q (x2, y2)
Origin

• If you choose a different coordinate frame, you
  will get a different result

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Vector and Affine Spaces

• Vector space
• Includes vectors and related operations
• No points
• Affine space
• Superset of vector space
• Includes vectors, points, and related operations

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Points and Vectors
Point q
vector (q - p)
Point p

• A point is a position specified with coordinate
  values.
• A vector is specified as the difference between
  two points.
• If an origin is specified, then a point can be
  represented by a vector from the origin.
• But, a point is still not a vector in
  coordinate-free concepts.

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Vector spaces

• A vector space consists of
• Set of vectors, together with
• Two operations addition of vectors and
  multiplication of vectors by scalar numbers
• A linear combination of vectors is also a vector

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Affine Spaces

• An affine space consists of
• Set of points, an associated vector space, and
• Two operations the difference between two points
  and the addition of a vector to a point

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Addition
p w
u v
w
v
u
p
u v is a vector
p w is a point
u, v, w vectors p, q points
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Subtraction
p
p - w
p - q
-w
u - v
u
v
q
p
u - v is a vector
p - q is a vector
p - w is a point
u, v, w vectors p, q points
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Linear Combination

• A linear space is spanned by a set of bases
• Any point in the space can be represented as a
  linear combination of bases

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Affine Combination
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General position

• "We assume that the points (lines, hyperplanes,.
  . . ) are in general position."
• No "unlikely coincidences" happen in the
  considered configuration.
• No three randomly chosen points are collinear.
• Points in lRd in general position, we assume
  similarly that no unnecessary affine dependencies
  exist No kltd1 points lie in a common
  (k-2)-flat.
• For lines in the plane in general position, we
  postulate that no 3 lines have a common point and
  no 2 are parallel.

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Convexity
A set S is convex if for any pair of points p,q ?
S we have pq ? S.
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Convex Hulls Equivalent definitions

• The intersection of all covex sets that contains
  P
• The intersection of all halfspaces that contains
  P.
• The union of all triangles determined by points
  in P.
• All convex combinations of points in P.

P here is a set of input points
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Convex hulls
Extreme point Int angle lt pi
p6
p9
p5
p7
p12
p4
p11
p1
p8
p2
p0
Extreme edge Supports the point set
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Caratheodory's theorem
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Separation theorem

• Let C, D?Rd be convex sets with CnDØ. Then there
  exist a hyperplane h such that C lies in one of
  the closed half-spaces determined by h, and D
  lies in the opposite closed half-space.
• In other words, there exists a unit vector a?Rd
  and a number b?R such that for all x?C we have
  lta, xgtb, and for all x?D we have lta, xgtb.
• If C and D are closed and at least one of them is
  bounded, they can be separated strictly in such
  a way that CnhDnhØ.

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Example for separation
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Sketch of proof

• We will assume that C and D are compact (i.e.,
  closed and bounded). The cartesian product C x
  D?R2d is a compact set too.
• Let us consider the function f (x, y)?x-y ,
  when
• (x, y) ? C x D.
• f attains its minimum, so there exist two points
  a?C and b?D such that a-b is the possible
  minimum.
• The hyperplane h perpendicular to the segment ab
  and passing through its midpoint will be the one
  that we are searching for.
• From elementary geometric reasoning, it is easily
  seen that h indeed separates the sets C and D.

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Farkas lemma

• For every d x n real matrix A, exactly one of the
  following cases occurs
• There exists an x?Rn such that Ax0 and xgt0
• There exists a y?Rd such that yT Alt0. Thus, if we
  multiply j-th equation in the system Ax0 by yi
  and add these equations together, we obtain an
  equation that obviously has no nontrivial
  nonnegative solution, since all the coefficients
  on the left-hand sides are strictly negative,
  while the right-hand side is 0.

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Proof of Farkas lemma

• Another version of the separation theorem.
• V?Rd be the set of n points given bye the column
  vectors of the matrix A.
• Two cases
• 0?conv(V)
• 0 is a convex combination of the points of V.
• The coefficients of this convex combination
  determine a nontrivial nonnegative solution to
  Ax0
• 0?conv(V)
• Exist hyperplane strictly separation V from 0,
  i.e., a unit vector y?Rn such that lty, vgt lt lty,
  0gt 0 for each v?V.

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Radons lemma

• Let A be a set of d2 points in Rd. Then there
  exist two disjoint subsets A1, A2?A such that
  conv(A1) n conv(A2)?Ø
• A point x ? conv(A1) n conv(A2) is called a Radon
  point of A.
• (A1, A2) is called Radon partition of A.

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Hellys theorem

• Let C1, C2, , Cn be convex sets in Rd, nd1.
  Suppose that the intersection of every d1 of
  these sets is nonempty. Then the intersection of
  all the Ci is nonempty.

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Proof of Hellys theorem

• Using Radons lemma.
• For a fixed d, we proceed by induction on n.
• The case nd1 is clear.
• So we suppose that n d2 and the statement of
  Hellys theorem holds for smaller n.
• nd2 is crucial case the result for larger n
  follows by a simple reduction.
• Suppose C1, C2, , Cn satisfying the assumption.
• If we leave out any one of these sets, the
  remaining sets have a nonempty intersection by
  the inductive assumption.
• Fix a point ai ? ?i?jCj and consider the points
  a1,a2, , ad2
• By Radons lemma, there exist disjoint index sets
  I1, I2 ?1, 2, , d2 such that

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Example to Hellys theorem
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Continue proof of Hellys theorem

• Consider i?1, 2, , n, then i?I1 or i?I2
• If i?I1 then each aj with j ? I1 lies in Ci and
  so x?conv(aj j ? I1)?Ci
• If i?I2 then each aj with j ? I2 lies in Ci and
  so x?conv(aj j ? I2)?Ci
• Therefore x ? ni1nCi

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Infinite version of Hellys theorem

• Let C be an arbitrary infinite family of compact
  convex sets in Rd such that any d1 of the sets
  have a nonempty intersection. Then all the sets
  of C have a nonempty intersection.
• Proof
• Any finite subfamily of C has a nonempty
  intersection. By a basic property of compactness,
  if we have an arbitrary family of compact sets
  such that each of its finite subfamilies has a
  nonempty intersection, then the entire family has
  a nonempty intersection.

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Centerpoint

• Definition 1 Let X be an n-point set in Rd. A
  point x ? Rd is called a centerpoint of X if each
  closed half-space containing x contains at least
  n/(d1) points of X.
• Definition 2 x is a centerpoint of X if and only
  if it lies in each open half space ? such that
  X??gtdn/(d1).

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Centerpoint theorem

• Each finite point set in Rd has at least one
  centerpoint.
• Proof
• Use Hellys theorem to conclude that all these
  open half-spaces intersect.
• But we have infinitely many half-spaces ? which
  are unbound and open.
• Consider the compact convex set conv(X??)??

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Centerpoint theorem(2)

• Run ? through all open-spaces with X??gtdn/(d1)
• We obtain a family C of compact convex sets.
• Each Ci contains more than dn/(d1) points of X.
• Intersection of any d1 Ci contains at least one
  point of X.
• The family C consists of finitely many distinct
  sets.(since X has finitely many distinct
  subsets).
• By Hellys theorem ?C?Ø, then each point in this
  intersection is a centerpoint.

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Ham-sandwich theorem

• Every d finite sets in Rd can be simultaneously
  bisected by a hyperplane. A hyperplane h bisects
  a finite set A if each of the open half-spaces
  defined by h contains at most ?A/2? points of A.

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Center transversal theorem

• Let 1kd and let A1, A2, , Ak be finite point
  sets in Rd. Then there exists a (k-1)-flat f such
  that for every hyperplane h containing f, both
  the closed half-spaces defined by h contain at
  least Ai/(d-k2) points of Ai i1, 2, , k.
• For kd its ham-sandwich theorem.
• For k1 its the centerpoint theorem.