On some centers of triangles

This post saves a post deleted from mathoverflow:
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We try to go further from https://mathoverflow.net/questions/484443/on-3-centers-of-triangles and record more questions.

Given any point P in a triangular region T, we denote by $d_1$, $d_2$ and $d_3$ the perpendicular distances from P to the 3 sides of T. We denote by $D_1$, $D_2$, $D_3$, the distances from P to the vertices of T.

It is known that the position of P that minimizes the Arithmetic Mean of the $D_1$, $D_2$, $D_3$ is the Torricelli point. But I don't know answers to the following:

1. Given any T how does one characterize the location of P within T that minimizes the arithmetic mean of $d_1$, $d_2$ and $d_3$?

2. Which position of P inside T maximizes the Geometric Mean of $d_1$, $d_2$ and $d_3$? Which P maximizes the GM of the D's?

3. Which P maximizes the Harmonic Mean of $d_1$, $d_2$ and $d_3$? Which P maximizes the HM of the D's?