Proving the Convexity of a Disk: A Step-by-Step Guide

Understanding the concept of convexity in a Euclidean space is crucial for many fields, including geometry, optimization, and computer science. In this article, we will explore how to prove that the interior of a circle, known as a disk, is convex. This proof is a fundamental concept that will be broken down into clear, manageable steps, making it accessible for learners of all levels.

Definition of Convexity

A set $S$ in a Euclidean space is defined to be convex if for any two points $x, y in S$, the line segment connecting $x$ and $y$ is also entirely contained within $S$. We will use this definition to prove the convexity of a disk.

Steps to Prove Convexity of a Disk

1. Define the Disk

Let $D$ be a disk with radius $r$ centered at the origin, defined as:

$D { (x, y) in mathbb{R}^2 : x^2 y^2 leq r^2 }$

2. Choose Two Points

Let $A (x_1, y_1)$ and $B (x_2, y_2)$ be any two arbitrary points in the disk $D$. This means that:

$x_1^2 y_1^2 leq r^2$ and $x_2^2 y_2^2 leq r^2$

3. Parameterize the Line Segment

The line segment connecting points $A$ and $B$ can be parameterized as:

$P_t (1 - t)A tB (1 - t)(x_1, y_1) t(x_2, y_2)$

for $t in [0, 1]$.

4. Show that the Line Segment is Inside the Disk

To show that the line segment is inside the disk, we must demonstrate that for all $t in [0, 1]$, the point $P_t$ is in $D$, which means:

$(1 - t)(x_1, y_1) t(x_2, y_2)$ is in $D$.

This can be derived as follows:

The squared distance from the origin to $P_t$ is:

$P_t^2 (1 - t)^2(x_1^2 y_1^2) t^2(x_2^2 y_2^2) 2t(1 - t)x_1x_2 2t(1 - t)y_1y_2$

Since $x_1^2 y_1^2 leq r^2$ and $x_2^2 y_2^2 leq r^2$, we have:

$P_t^2 leq (1 - t)^2r^2 t^2r^2$

$P_t^2 leq r^2[(1 - t)^2 t^2]$

$P_t^2 leq r^2(1 - 2t t^2 t^2)$

$P_t^2 leq r^2(1 - 2t 2t^2)$

Note that $1 - 2t 2t^2 leq 1$ for all $t in [0, 1]$, so:

$P_t^2 leq r^2$

Therefore, the point $P_t$ is indeed in the disk $D$ for all $t in [0, 1]$.

Intuitive Proof Using the Triangle Inequality

Intuitively, this result follows from the fact that any point on the line segment between two points within a circle remains within the circle. Consider two points $A$ and $B$ inside a circle centered at $C$. Any point $D$ on the line segment $AB$ is inside the circle because the distance from $D$ to $C$ is less than or equal to the radius of the circle.

Conclusion

The disk is proven to be a convex set. By demonstrating that any line segment connecting two points within the disk lies entirely within the disk, we have shown that the disk satisfies the definition of convexity.

Thus, we have rigorously proven that the interior of a circle is convex, providing a solid mathematical foundation for further exploration in geometry and related fields.