After searching on Google for a while, I cannot seem to find an algebraic proof that a disk is a convex set. Intuitively, this seems obvious: if you take any two points $x, y$ in a disk, then the line from $x$ to $y$ is clearly contained in the disk.
More specifically, let $D \subset \mathbb{R}^2$ be an origin-centered disk of radius $r$. Let $x = (x_1, x_2), y = (y_1, y_2) \in C$. We wish to show that for any $\lambda \in [0, 1]$, the point $z$ given by
$$
z = \lambda x + (1 – \lambda) y = (\lambda x_1 + (1 – \lambda y_1), \lambda x_2 + (1 – \lambda) y_2)
$$
also lies in $D$. By definition, we have $\|x\|_2^2 \leq r^2$ and $\|y\|_2^2 \leq r^2$. Using this information, I tried to show that $\|z\|_2^2 \leq r^2$ as follows:
\begin{align*}
(\lambda x_1 + (1 – \lambda y_1))^2 + (\lambda x_2 + (1 – \lambda) y_2)^2
&= \lambda^2 x_1^2 + 2 \lambda (1 – \lambda) x_1 y_1 + (1 – \lambda)^2 y_1^2 + \\
&\phantom{50} \lambda_2 x_2^2 + 2 \lambda (1 – \lambda) x_2 y_2 + (1 – \lambda)^2 y_2^2 \\
&= \lambda^2 (x_1^2 + x_2^2) + 2 \lambda (1 – \lambda) (x_1 y_1 + x_2 y_2) + (1 – \lambda)^2 (y_1^2 + y_2^2).
\end{align*}
Unfortunately, I'm stuck at this point — I can't see a clean way to factor this expression and bound it above by $r^2$. I feel like I am missing obvious, but at this time I cannot determine what that is. Is there a clean proof of this fact that follows from this reasoning?
Edit: Solution using Cauchy-Schwarz Inequality
Even though I used the notation for norms, it didn't occur to me to just use the triangle inequality, as Dustan Levenstein suggested. The solution would then proceed as follows:
\begin{align*}
\| \lambda x + (1 – \lambda) y \|_2
&\leq \lambda \| x \|_2 + (1 – \lambda) \| y \|_2 \\
&\leq \lambda r + (1 – \lambda) r = r.
\end{align*}
It follows that $\| z \|_2^2 \leq r^2$, as desired. I'm still going to leave the question up, in case there's a solution that does not rely on the Cauchy-Schwarz inequality.
Best Answer
Equation of a disk: $x^2+y^2≤r^2$ (r is the radius).
Let’s say 2 points in the disk are $(x_1,y_1)$ and $(x_2,y_2)$.
Now we have to show that any point on the line segment joined by these 2 points i.e. $(tx_1+(1-t)x_2, ty_1+(1-t)y_2)$ also satisfies the disk equation, where $1 \leq t \leq 0$.
$(tx_1+(1-t)x_2)^2 + (ty_1+(1-t)y_2)^2$
$= t^2 x_1^2+t^2 y_1^2+(1-t)^2 x_2^2+(1-t)^2 y_2^2 + 2t(1-t)[x_1x_2+y_1y_2]$
$= t^2(x_1^2+y_1^2)+(1-t)^2(x_2^2+y_2^2) + t(1-t)(2x_1x_2 + 2y_1y_2)$
$\leq t^2 r^2+ (1-t)^2 r^2 + t(1-t)(x_1^2+x_2^2+y_1^2+y_2^2)$, because $2xy≤x^2+y^2$,prove it yourself.
$ \leq t^2r^2+ (1-t)^2r^2 + t(1-t)(r^2+r^2) = r^2 [t^2+(1-t)^2 + 2t(1-t)]$
$= r^2$
Hence proven disk is convex.