Is it a convex set?

Question

I have the following set

$$A = \{ (x,y) \in \Bbb R^{2} : \log x + y^{2}\ge 1, x \ge 1, y \ge 0 \}$$

and I need to know if it's convex or not. I tried to have a look at this function $-\log x-y^{2}$, but the Hessian matrix is indefinite and I don't know what to do else.

Answer 1

$(e,0)$ is in the set.

$(1,1)$ is also in the set.

The corresponding straight line connecting the two points are

$$y-1 = \frac{-1}{e-1}(x-1)$$

A point on that line segment is $x=2$ and $y=1-\frac1{e-1}$.

$$\ln 2+(1-\frac1{e-1})^2=\ln 2 + 1 - \frac2{e-1}+\frac1{(e-1)^2} \approx 0.693 + 1 -1.164+0.339 < 1$$

Answer 2

HINT

Since $y\ge 0$ in your set, the first equation implies $$y^2 \ge 1 - \ln x \implies y \ge (1 - \ln x)^{1/2}.$$

Can you check that $f(x) = (1 - \ln x)^{1/2}$ is convex or concave? Note also this clearly implies you must have $\ln x \le 1 \iff x \le e$.