To explain normal cones in simpler terms, let’s first start with an example.

Imagine you’re standing on a smooth surface, like a hill. If you’re at a certain point on the hill, you might want to know the direction of the steepest descent or the steepest slope. That’s easy when the surface is smooth. But what if you’re at the edge of a cliff or a sharp corner? There, it’s not as simple to define which way is “down” or “up” because the surface isn’t smooth anymore.

This is where normal cones come into play.

What is a Normal Cone?

In geometry and optimization, a normal cone is a mathematical object that helps us describe how a surface or shape behaves around edges or corners, where things get more complicated. The normal cone tells us which directions we can move away from that point without going inside the shape.

Let’s break it down:

• Think of a point on the surface of a shape.

• The normal cone at that point shows all the possible directions you can move outward from that point without touching the inside of the shape.

Example with a Square

Imagine a flat square on a piece of paper. If you’re at a point inside the square, the steepest descent (or the direction you’d move “outward”) is easy to figure out – just go straight up, down, left, or right. But if you’re at a corner of the square, it’s harder to tell which way to move. At the corner, the normal cone is the space between the two edges meeting at that corner. This cone tells us that the directions we can move outward are along or between those edges.

Key Takeaway

A normal cone helps describe the directions you can “push” outward from a point when you’re at a sharp corner or edge of a shape. It’s a way of dealing with points where the shape isn’t smooth, and it helps in solving optimization problems or understanding the behavior of the shape around those complex points.