The interactive film is in the book's Companion

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Norms as Cones: the Unit Ball Is a Slice

Chapter 1 · Mathematics for AI and Machine Learning · Beginner · 1:42

An interactive live film: every norm drawn as a height over the plane is a cone, and its slice at height one is the unit ball: circle, diamond, square, tilted ellipse. Find the p whose ball passes through (0.8, 0.8).

Narration

A norm gives every vector a length. Draw that length as a height above the plane, and each norm becomes a surface.

Double a vector and its length doubles, so every such surface is a cone with its tip at the origin. Slice it at height one, and the cut is the unit ball: all vectors of length one.

The rest of the narration is in the book's Companion.

More from Chapter 1

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