This is based off of my lengthy answer here regarding the location of the Bimah in a shul. As a result of that research, I was left with one question.
In that answer, I quote R' Moshe Feinstein in Igros Moshe OC 1:42, who says:
בדבר הבימה אין קפידא שיהיה באמצע בית הכנסת ממש, וכמעט שעוד עדיף שיהיה קרוב יותר לפנים, דברוב פעמים אין המתפללין ממלאין כל ביהכ״נ ועומדים רובם יותר קרוב לפנים וממילא צריכה הבימה להיות באמצע עמידת המתפללין כיון שעיקר הטעם הוא כדי שישמעו העם קה״ת.
Regarding the Bimah: We aren’t particular that it should be literally in the middle of the shul, and if anything, it’s better that it should be closer to the front, since many times the congregants don’t fill the entire shul, and the majority of them stand toward the front, and it comes out that the Bimah needs to be in the middle of where the congregants are standing, since the main reason is that the congregation can hear the reading of the Torah.
And I inferred from there:
While I don’t know that the logic of R’ Moshe would apply to every shul that it should be toward the front rather than the back, the underlying logic would remain: wherever congregants tend to stand when the shul is only partially full, the center should be where the Bimah should go.
Based on the above, what defines the "center" of the congregation? Let's say that 90% of the congregation is in the front, and 10% is in the back, with a wide gap in between them. If I were to draw a shape that perfectly encompasses all congregants, the center of that shape would roughly approximate the actual center of the room. On the other hand, 90% of the congregation is in the front, so perhaps the Bimah should be further toward the front, in the mean center of the population.
I am asking specifically according to this understanding of R' Moshe. I don't want answers according to other opinions. I will accept answers that can prove my assumption wrong, though I would rather an answer that can demonstrate from a different teshuvah of R' Moshe if he addresses what he means by the center.