Lines, Hyperplanes, and Convex Sets

Chapter 9 gave ${\mathbb{R}}^n$ a distance. This chapter asks how much of the familiar geometry that distance already decides. A great deal: lines and segments, spheres and balls are all sets cut out by distances, and “$x$ lies between $p$ and $q$” is a statement about three distances and nothing else. One linear equation $⟨a,x⟩=c$ cuts out a hyperplane, and the distance from a point to it has a formula. A set is convex when it contains the segment between any two of its points. The smallest convex set containing a given set is the set of its convex combinations.

Analysis uses all of this from its first chapters on. The mean value theorem moves along a segment, and a ball is the set of points where an estimate holds. In a convex set, an argument made along one segment works between any two points.

What you need.

• Vectors, the dot product, the norm and the distance, from Chapter 9, Sections 9.2 to 9.4 (Definition 9.4.1).
• The triangle inequality and its equality case, from Section 9.5 (Theorem 9.5.3).
• Projection onto a line and the closest-point theorem, from Section 9.6 (Theorem 9.6.1).
• The rules of an ordered field and square roots in $\mathbb{R}$, from Number Systems: From $\mathbb{N}$ to $\mathbb{R}$ (Number Systems: From $\mathbb{N}$ to $\mathbb{R}$, Proposition 6.1.4).
• Finite sums and induction, from Chapter 1 and Sets and Functions, for convex combinations of many points (Sets and Functions, Theorem 3.2.1).

Time. About 15 hours if you read the main text and do every A exercise and one B and one C exercise per section; about 20 hours for everything.

Roadmap

• Lines and Segments. Lines $p+\mathbb{R}v$, segments $[p,q]$, and the theorem that $x$ lies on $[p,q]$ exactly when $\Vert p-x\Vert +\Vert x-q\Vert =\Vert p-q\Vert$ (Theorem 10.1.4).
• Hyperplanes and Nearest Points. Hyperplanes $⟨a,x⟩=c$, the nearest point on a line or a hyperplane with the distance to it (Proposition 10.2.4), and how a line meets a sphere.
• Convex Sets and Hulls. Convex sets, which include balls (Proposition 10.3.1), convex combinations, and the convex hull of a set as the set of convex combinations of its points.