Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.)
Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, where $p > 1$, $p \neq 2$.
Is it correct that $E$ and $S$ have at most 8 intersections?
More exactly what I care about: is it true that $E$ and $S$ have at most 2 intersections in the nonnegative quadrant?
It seems relatively clear from drawing pictures, but I couldn't quite get the natural convexity argument to work.