1.3 Operations preserving convexity

Convex sets are fundamental in geometry, defined by line segments between any two points staying within the set. This property extends to intersections, affine transformations, and Minkowski sums, forming a powerful toolkit for analyzing geometric shapes.

Understanding how convexity behaves under various operations is crucial. While intersections and affine transformations preserve convexity, unions and complements generally don't. These insights help in manipulating and analyzing complex geometric structures in higher dimensions.

Set Operations and Convexity

Convexity of set intersections

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• Definition of convex sets encompasses any set where line segments connecting two points within the set lie entirely within the set (spheres, cubes)
• Intersection of sets comprises points belonging to all sets being intersected, forming a common region (overlapping area of Venn diagram)
• Proof strategy involves considering arbitrary points in intersection and demonstrating line segment between them remains within intersection
• Key steps in proof:
  1. Select two points $x$ and $y$ in intersection
  2. Consider point $z = \lambda x + (1-\lambda)y$ where $0 \leq \lambda \leq 1$ on line segment
  3. Demonstrate $z$ exists in each intersecting set
  4. Conclude $z$ belongs to intersection, establishing convexity

Convexity under affine transformations

• Affine transformations combine linear transformations with translations, expressed as $T(x) = Ax + b$ (rotation, scaling, shearing)
• Properties of affine transformations maintain collinearity and preserve ratios of distances between points
• Proof approach starts with convex set $S$, applies affine transformation $T$, and shows resulting $T(S)$ remains convex
• Key steps:
  1. Consider two points in $T(S)$
  2. Identify their preimages in $S$
  3. Utilize convexity of $S$ and affine transformation properties
  4. Establish convexity of $T(S)$

Convexity in Minkowski sums

• Minkowski sum defined as $A + B = \{a + b : a \in A, b \in B\}$, geometrically interpreted as sweeping one set over another
• Scalar multiplication of sets defined as $\alpha A = \{\alpha a : a \in A\}$, affects set shape and size (doubling, halving)
• Convexity of Minkowski sum proof involves showing $A + B$ remains convex when $A$ and $B$ are convex
• Convexity of scalar multiples:
  • Proven for positive scalars
  • Zero scalar results in single point (origin)
  • Negative scalars flip set orientation

Convexity of set operations

• Union of convex sets not always convex, counterexample: union of two separate circles
• Complement of convex set generally not convex, exception: half-spaces remain convex
• Difference of convex sets lacks guaranteed convexity, preserved in some cases (nested convex sets) but not others (overlapping convex sets)
• Convex hull represents smallest convex set containing given points or sets, always convex by definition
• Projection of convex sets onto lower-dimensional subspaces preserves convexity (3D sphere projected onto 2D plane becomes circle)