Pick fixed $a=(a_1,a_2,\dots,a_d)\in\{\pm1\}^d$.
Consider map $F_a:\underbrace{\Bbb R^n\times\dots\times \Bbb R^n}_d\rightarrow\Bbb R^n$ given by $F(x_1,\dots,x_d)=\sum_{i=1}^da_ix_i$.
Denote $S_n\subset\Bbb R^n$ as unit sphere at origin, $B_r\subset\Bbb R^n$ to be closed ball (not sphere) of radius $r$ at origin.
Fix $K\in(0,1)$.
Algebraically or geometrically, what does the space $$F_a^{-1}\big(B_K\big)\bigcap\underbrace{S_n\times\dots\times S_n}_d=F_a^{-1}\big(B_K\big)\bigcap S_n^d$$ look like?
What is its measure in $S_n^d$?
Is there a $K\in(0,1)$ such that $\bigcup_{a\in\{\pm1\}^d} \Big(F_a^{-1}\big(B_K\big)\bigcap S_n^d\Big)\subsetneq S_n^d$?