Convexity of set A = {(x,y) in R^2 | x^4+y^4 =< 1, x>=0

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SUMMARY

The set A = {(x,y) in R^2 | x^4+y^4 <= 1, x>=0, y>=0} is confirmed to be convex based on the properties of convex functions. Specifically, the lower contour set defined by the inequality x^4 + y^4 <= 1 is convex, as it is derived from a weakly convex function. This conclusion is drawn from the definition of convexity in relation to the function's lower contour sets.

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peteryellow
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How can I show that the set A = {(x,y) in R^2 | x^4+y^4 =< 1, x>=0 y>=0} is convex.
 
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I am sure it can be shown by definition, but I propose an easy way (not rigorous though)

The function is (weakly) convex
The lower contour set (=<1) of a convect function is convex.
 

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