For example, I attached the figure. The first figure located at the left showed that the red nonlinear curve is always in the black inequality region. However, in the right figure, the red nonlinear curve does not always exist in the black inequality area.
Suppose that the black areas are from the second order polynomials' inequality. red line could be nonlinear curve or curved plane(black area is always higher dimension). Would there be any way to determine the red one in the black area?