Let $x,y, u, v \in \mathbb{R}^n$. Under what condition on $x,y, u, v$ we have the following: $$ \langle v-u, y-x \rangle \geq \alpha ||y-x||^2 \to ||v-u|| \leq \beta||y-x|| $$ for some $\alpha, \beta>0$.
My try: The assumption can be written as the following: $$ \langle v-\alpha y -(u-\alpha x), y-x \rangle \geq 0 $$ but how that can help is unclear to me. Also, when $x\neq y$, we can get $||v-u|| \geq \alpha||y-x||$ which is the opposite of what I am looking for.
Is there any way to have another inner product including $x, y, u, v$ to get $||v-u|| \leq \beta||y-x||$?