3 problems

Slice double-point conjecture. If $d\in\{4,5\}$, then, outside a single null set,

Negative-slice conjecture. Outside a single null set, the slice need not be a finite set for every $s>0$; indeed, there may exist a non-trivial measure function $\phi$ such that

Let $f_0$ have $\partial^d f_0(\boldsymbol{x})/\partial x_1\cdots\partial x_d$ continuous in a neighborhood of $\boldsymbol{x}_0$, with … Let…