Definition: Here a neighborhood of the point c → ∈ R n is the set...

Ryan Robertson

Ryan Robertson

Answered

2022-07-08

Definition: Here a neighborhood of the point c R n is the set N r ( c ) = { x R n : x c 2 < r } for some r > 0, and a subset U R n is said to be convex if x , y U , t [ 0 , 1 ] implies t y + ( 1 t ) x U.

My Attempt: Suppose x , y N r ( c ) and t [ 0 , 1 ], we then have x c 2 < r and y c 2 < r. Now,
t y + ( 1 t ) x c 2 = t y t x + x c 2 t y t x 2 + x c 2 < | t | y x 2 + r | t | ( y c 2 + c x 2 ) + r < | t | ( r + r ) + r = ( 1 + 2 | t | ) r
Screeching halt.

My Question: How should I make t y + ( 1 t ) x c 2 less than r? My intuition is that I need the Cauchy-Schwarz inequality; but I am unsure where to apply it. Any hint would be greatly appreciated.

Answer & Explanation

Miles Mueller

Miles Mueller

Expert

2022-07-09Added 11 answers

t y + ( 1 t ) x c 2 = t y + ( 1 t ) x t c ( 1 t ) c 2 = t ( y c ) + ( 1 t ) ( x c ) 2 t ( y c ) 2 + ( 1 t ) ( x c ) 2 = t y c 2 + ( 1 t ) x c 2 < t r + ( 1 t ) r = r .
Concerning < note that not both t and 1 t can be zero.

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