Using concexity property of norm function

Question

Let $(X,\|.\|)$ be a Banach space. Let $x\neq 0$ and $y\neq 0$ be in $X$ such that there exists a non zero $\lambda_0\in\mathbb{R}$, $\lambda_0$ to be<0, such that $\|x+\lambda_0y\|<\|x\|$. Can anyone tell me how by convexity of the norm function $\|x+\lambda y\|<\|x\|$ for $\lambda \ in [\lambda_0, 0).$?

Answer 1

Set $y_0=\lambda_0y$, then $\|x+y_0\|<\|x\|$. By triangle inequality (or the convexity property as you mention) we have, for $\gamma\in(0,1]$, $$ \|x+\gamma y_0\|=\|\gamma(x+y_0)+(1-\gamma) x\|\leq\gamma\|x+y_0\|+(1-\gamma)\|x\|<\|x\|, $$ that is to say, for $\lambda\in\lambda_0(0,1]$, we have $$ \|x+\lambda y\|<\|x\|. $$