High-dimensional geometry is full of surprises. For example, nearly all the area of a high-dimensional sphere is near the equator, and by symmetry it doesn’t matter which equator you take.
Here’s another surprise: corners stick out more in high dimensions. Hypercubes, for example, become pointier as dimension increases.
How might we quantify this? Think of a pyramid and a flag pole. If you imagine a ball centered at the top of a pyramid, a fair proportion of the volume of the ball contains part of the pyramid. But if you do the same for a flag pole, only a small proportion of the ball contains pole; nearly all the volume of the ball is air.
So one way to quantify how pointy a corner is would be to look at a neighborhood of the corner and measure how much of the neighborhood intersects the solid that the corner is part of. The less volume, the pointier the corner.
Consider a unit square. Put a disk of radius r at a corner, with r < 1. One quarter of that disk will be inside the square. So the proportion of the square near a particular corner is πr²/4, and the proportion of the square near any corner is πr².
Now do the analogous exercise for a unit cube. Look at a ball of radius r < 1 centered at a corner. One eighth of the volume of that ball contains part of the cube. The proportion of cube’s volume located within a distance r of a particular corner is πr³/6, and the proportion located within a distance r of any corner is 4πr³/3.
The corner of a cube sticks out a little more than the corner of a square. 79% of a square is within a distance 0.5 of a corner, while the proportion is 52% for a cube. In that sense, the corners of a cube stick out a little more than the corners of a square.
Now let’s look at a hypercube of dimension n. Let V be the volume of an n-dimensional ball of radius r < 1. The proportion of the hypercube’s volume located within a distance r of a particular corner is V / 2n and the proportion located with a distance r of any corner is simply V.
The equation for the volume V is
If we fix r and let n vary, this function decreases rapidly as n increases.
Saying that corners stick out more in high dimensions is a corollary of the more widely known fact that a ball in a box takes up less and less volume as the dimension of the ball and the box increase.
Let’s set r = 1/2 and plot how the volume of a ball varies with dimension n.
You could think of this as the volume of a ball sitting inside a unit hypercube, or more relevant to the topic of this post, the proportion of the volume of the hypercube located with a distance 1/2 of a corner.