An exercise in analysis about normed vector spaces

Question

I have found the following exercise :

Let $E$ be a normed vector space and $x,y\in E$ be non-zero vectors.

1. Show that $\left\Vert x-y\right\Vert \geq \frac{1}{4}\left( \left\Vert x\right\Vert +\left\Vert y\right\Vert \right) \left\Vert \frac{x}{\left\Vert x\right\Vert }-\frac{y}{\left\Vert y\right\Vert }\right\Vert $

2. Show that the constant $\frac{1}{4}$ can not be replaced with a greater one.

I was able to prove the above inequality but concerning the constant $\frac{1% }{4}$ I have shawn that it can be replaced with $\frac{1}{2}$ in case of Hilbert spaces.

So my question is : is there an example of a normed verctor space where the constant $\frac{1}{4}$ can not be improved ?

Thank you !