There are two main kinds of arguments against abortion: Those based on the idea that we begin existing at conception and those based on the idea that personhood begins at conception.
One of the main objections to thinking that our existence begins at conception is the incredulous stare: How can that single cell be me?!
Here my recent geometrical observations about how I will be very small in almost every reference frames become relevant. Exactly the same argument establishes that in almost every reference frame, I start out really small. In almost every reference frame, I start out less than a nanometer in size (any non-zero size can be substituted here), and hence much smaller than a single cell.
Thus, it seems we are simply stuck with a counterintuitive result about what we are like at our beginning. Even if we don’t begin at conception, in almost all reference frames we begin as something much smaller than a single cell.
Can the geometrical observations show that personhood begins really small, too, and thereby undercut the incredulous stare at the idea that a single cell is a person?
Now, if we are essentially persons, given that by the previous argument we begin smaller than a cell, then indeed something smaller than a cell is a person.
So the remaining case to consider is views on which we are only accidentally persons, and we pre-exist our personhood. A typical view in this family will say that we are animals that come into existence at conception or implantation, and that about 1.5-2 years after our beginning, we come to have the property of personhood.
In the previous argument, I looked at the set K of all the spacetime locations of my body, and it followed that for almost every reference frame F, there was a time t in F and near my beginning such that the t-slice of me was really tiny. The obvious analog is to look at the set K* of all spacetime locations of my personal body—i.e., of my body at times at which I am a person—and repeat the argument. The problem with this move is that whether a spacetime location is within my body is intuitively independent of reference frame, but whether a spacetime location is within my personal body could more plausibly depend on the reference frame, if my 4D personal body is not all of my 4D body.
So at this point, I don’t have a version of my smallness argument against the view that to be a person I have to be big, when that view is coupled with the idea that I can exist without being a person.