Let $t ∈ [−1,1]$, describe the set $S(t) := B_n ∩\lbrace x ∈ \Bbb R^n : x_n = t\rbrace $ geometrically as a subset of $\Bbb R^{n−1}$. Here $B_n$ denotes the $n$-dimensional unit ball.
That is, we need to identified the points of form $(x_1,x_2,...,x_{n−1},t) ∈ \Bbb R^n$ with the corresponding $(x_1,x_2,...,x_{n−1}) ∈ \Bbb R^{n−1}$ in the obvious way.
It seems to me that everything in high-dimensional space are quite special so I am worrying about if there exist some obvious way to do that.
So may I please ask if there are some known formula about this? Any help or reference would be appreciate. Or should we construct some formula for it? May I please ask for an answer?
Also I am considering about the volume: Once we have identified the set, may I please ask for the $(n−1)$-dimensional volume of the set $S(t)$? How can we express it in terms of $V_{n−1}$?
Thanks a lot!
Presumably, we have $|t|<1$. Else, the intersection is either empty, or a single point (namely $0,\ldots,0,t)$).
Once that is true, we are left with the inequality
$$x_1^2+\cdots+x_{n-1}^2\leq1-t^2$$
This is an $n-1$-dimensional ball, but not a unit ball (unless $t=0$). Its radius is, in fact, $\sqrt{1-t^2}$.
As for its volume, do you know the formula for the volume of a $k$-dimensional ball of radius $r$?