Let $y \in \mathbb{R}^{n}$ and define the set $$ K(y) = \{ x \in \mathbb{R}^n: \|y + \alpha x\| < \|y\|, \alpha > 0 \} $$
I want to express this set in simpler terms so that I can better understand the nature of these vectors $x \in \mathbb{R}^{n}$.
Obviously, $\|y + \alpha x\| \leqslant \|y\| + \alpha\|x\|$ by the triangle inequality; however, I do not see how to put this together, or if it has any relevance to my problem.
Do any users have any insight into how I can express this relation inside of $K(y)$ more simply?
Yes. Multiplication by $\alpha$ scales (i.e., stretches or compresses and, possibly, reverses the direction of) the vector. So, $y + \alpha x$ reads as "the translate of $y$ by the scaling $\alpha x$ of $x$.
One ambiguity in your definition of $K(y)$ is whether it means "for some $\alpha > 0$" or "for all $\alpha > 0$"). Whatever the case, your $K(y)$ is the set of all $x$'s such that translations by $\alpha x$ do not go outside of the sphere of radius $||y||$. So, pick a $y$ in the 2-dimensional Euclidean space and compute its $K(y)$.