if $\|A - B\| > \|A - C\|$, is it then true that $\|A - \frac12(B+C)\| > \|A - C\|$

Question

Given that $\|A-B\| > \|A-C\|$ I can show that $\|A - B\| > \|A - 1/2(B+C)\|$:

Using the triangle inequality:

$$1/2(\|A - B\| + \|A - C\|) \geq \|A - 1/2(B+C)\|$$

and since $\|A-B\| > \|A-C\|$, we get:

$$\|A- B\| \geq \|A - 1/2(B+C)\|$$

However I cant prove the statement $\|A - 1/2(B+C)\| \geq \|A - C\|$. Intuitively it seems true and if I draw a few vectors it is true ...

Any help would be appreciated.

Answer 1

No, it's not true. A counter example is $A=0$, $B=2$ and $C=-1$ which results in $\frac{1}{2}\left(B+C\right)=\frac{1}{2}$, and thus, $\|A-C\|=1>\|A-\frac{1}{2}\left(B+C\right)\|=\frac{1}{2}$.