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I need to show that the following set is convex:

$$\{(x,y): \{||(x,3)|| \leq 5y, \: y \leq 1\} $$

So I rewrote the first constraint to $ 25y^2 - x^2 \geq 9$ and the minimum of this constraint is at $(x,y) = (0, 0.6)$

So I came up with the following:

Suppose we have $x = \langle x_1, x_2 \rangle , \: y=\langle y_1, y_2 \rangle \in S \text{ for } 0 \leq t \leq 1$

Then we get the following:

$$(1-t)x + ty = \langle (1-t)x_1 + ty_1, (1-t)x_2+ty_2 \rangle =\\ 25\left[(1-t)x_2+ty_2 \right]\left[(1-t)x_2+ty_2 \right] - [(1-t)x_1 + ty_1)((1-t)x_1+ty_1)=\\ 25[(1-t)^2x_2^2 + 2(1-t)tx_2y_2 + t^2y_2^2] - \;... \\ [(1-t)^2x_1^2 + 2(1-t)tx_1y_1 + t^2y_1^2] = \\ (1-t)^2 (25x_2^2 - x_1^2) + t^2 (25x_2^2 - x_1^2) \;+ \;... $$

I am a bit confused by the exponentials and as a result, the multiplications, e.g. $2(1-t)tx_2y_2$ . Am I doing this correctly or am I making a mistake?

I was also thinking about creating a composite function but I am not quite sure.

mathguy69
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    This set is not convex (note $(x, y) \in S$ implies $(-x, -y)\in S$, so $(0,0) \in S$ if $S$ is convex) – Arctic Char Nov 09 '20 at 15:10
  • but this is not an equation rather an inequality – Albus Dumbledore Nov 09 '20 at 15:23
  • While your reformulation is nonconvex, the original set you wrote is indeed convex. What tools can you use from your class to prove convexity? Do you need to prove it directly, or can you use the fact that a lower-level set of a convex function is convex? – Zim Jan 17 '24 at 10:45

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