dg.differential geometry - Convexity and Strong convexity of subsets of Surfaces

I believe your example is correct. As you are allowing $A$ closed the following is pretty similar. Choose a real number $N > 1.$ Take the smooth and analytic curve (write it as a level curve and check the gradient)

$$y^2 = x^3 - 3 N^2 x^2 + 3 N^4 x = left( frac{x}{4} right) left( (2 x - 3 N^2)^2 + 3 N^4 right)$$
Note that $$2 y y' = 3 x^2 - 6 N^2 x + 3 N^4 = 3 (x - N^2)^2.$$
Also $$4 y^3 y'' = 3 (x^4 - 4 N^2 x^3 + 6 N^4 x^2 - 3 N^8 ) = 3left((x-N^2)^4 + 4N^6(x-N^2)right).$$
Furthermore, when $$x = N^2, y = pm N^3.$$
Revolve this around the $x$-axis, making a simply connected surface. There is now a closed geodesic along $x = N^2,$ of circumference $2 pi N^3.$ The minimizing geodesic between
$( N^2, N^3, 0)$ and $( N^2, -N^3, 0)$ is the original curve in the plane $z=0,$ of length no larger than $2 N^2 + 2 N^3.$ The length of half the closed geodesic is $pi N^3,$ which is larger for large enough $N.$ So, as I did not say, we are taking $N > 2$ and
$$A = left{(x,y,z) in mathbb R^3 : y^2 + z^2 = x^3 - 3 N^2 x^2 + 3 N^4 x ; mbox{and} ; x leq N^2.right}$$

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