How to show that a set ##C=\{(x,y,z)\in\mathbb{R}^3:x\geq0,z\geq0,xz\geq y^2\}## is convex?(adsbygoogle = window.adsbygoogle || []).push({});

I tried a proof by contradiction: Assume that there exist ##c_1=(x_1,y_1,z_1),c_2=(x_2,y_2,z_2)\in C## and ##t\in(0,1)## such that ##tc_1+(1-t)c_2\notin C##.

For this to hold, one would have to have

$$(tx_1+(1-t)x_2)(tz_1+(1-t)z_2)<(ty_1+(1-t)y_2)^2$$ $$t^2x_1z_1+t(1-t)(x_1z_2+x_2z_1)+(1-t)^2x_2z_2<t^2y_1^2+2t(1-t)y_1y_2+(1-t)^2y_2^2.$$ We know that ##x_1z_1\geq y_1^2## and ##x_2z_2\geq y_2^2## but this doesn't seem to help me get any further than ##x_1z_2+x_2z_1<2y_1y_2##, from which I can't see a possible contradiction.

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# Proving convexity of a set

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