15. $z$ is a complex number, then the minimum value of $T=|z|+|z-2|+|z+\sqrt{3} i|$ is what?

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15. $T=|z|+|z \cdot 2|+|z+\sqrt{3} i|$ The minimum value is the minimum sum of distances from a point $P$ to points $A(0,0), B(2,0), C(0,-\sqrt{3})$ in the Cartesian coordinate system, hence $P$ is the Fermat point of $\triangle A B C$. Let $|P A|=x, |P B|=y, |P C|=z$.
So $\left\{\begin{array}{l}x^{2}+y^{2}-200 \operatorname{s} 120^{\circ} x y=2^{2} \\ x^{2}+z^{2}-2 \cos 120^{\circ} x z=(\sqrt{3})^{2} \\ y^{2}+z^{2}-2 \cos 120^{\circ} y z=2^{2}+(\sqrt{3})^{2}\end{array}\right.$

Also, $S_{\triangle A B C}=\frac{1}{2} \cdot 2 \cdot \sqrt{3}=S_{\triangle P A B}+S_{\triangle P A C}+S_{\triangle P A C}$
$$
=\frac{1}{2} \cdot \sin 120^{\circ}(x y+y z+x z)
$$

So $x y+y z+x z=4$.
So $2\left(x^{2}+y^{2}+z^{2}\right)+(x y+y z+x z)=14$.
So $x^{2}+y^{2}+z^{2}=5$.
So $(x+y+z)^{2}=13$.
So the minimum value of $T$ is $\sqrt{13}$.