What conditions should vector x satisfy so that ‖ [ x 2 + α x 1 , … , x n + α x 1 ] ‖ 2 is bounded by a constant?Suppose that x = [ x 1 , … , x n ] is a vector with norm less than or equal to one ‖ x ‖ 2 2 ≤ 1. Let α ∈ [ 0 , 1 ] and define the following vector y = [ x 2 + α x 1 , … , x n + α x 1 ]How can I find a non-trivial subset of x that, regardless of the value of α, would result in ‖ y ‖ 2 2 ≤ C where C is a constant that does not depend on n?The above is satisfied when x 1 = ⋯ = x n . For example, if x = [ 1 / n , … , 1 / n ] then, we have ‖ y ‖ 2 2 = ( 1 + α ) 2 n × n = ( 1 + α ) 2 ≤ 4. But I'm looking for a larger subset (or all x that satisfy the above conditions) than x 1 = ⋯ = x n

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What conditions should vector x satisfy so that   ‖  [      x    2    +  α      x    1    ,  …  ,      x    n    +  α      x    1    ]      ‖    2   is bounded by a constant?Suppose that   x  =  [      x    1    ,  …  ,      x    n    ] is a vector with norm less than or equal to one   ‖  x      ‖    2    2    ≤  1. Let   α  ∈  [  0  ,  1  ] and define the following vector  y  =  [      x    2    +  α      x    1    ,  …  ,      x    n    +  α      x    1    ]How can I find a non-trivial subset of x that, regardless of the value of α, would result in   ‖  y      ‖    2    2    ≤  C where C is a constant that does not depend on n?The above is satisfied when       x    1    =  ⋯  =      x    n  . For example, if   x  =  [  1      /        n    ,  …  ,  1      /        n    ] then, we have   ‖  y      ‖    2    2    =            (      1      +      α              )        2              n    ×  n  =  (  1  +  α      )    2    ≤  4. But I&#039;m looking for a larger subset (or all x that satisfy the above conditions) than       x    1    =  ⋯  =      x    n

Mario Kerr · Accepted Answer

You just need the condition       |        x    1        |    ≤  1      /        n   and you get   ‖  y      ‖    2    ≤  4. In fact,  ‖  y      ‖    2    =  [  (  n  −  1  )  α  −  1  −  2  α  ]      x    1    2    +  ∑      x    1    2    +  2  α      x    1    ∑      x    i  but  ∑      x    i    ≤  n      ∑          x      i      2              /        n    ≤      n  so  ‖  y      ‖    2    ≤  [  (  n  −  3  )  α  −  1  ]      x    1    2    +  1  +  2  α      x    1        n    ≤  4

What conditions should vector x satisfy so that norm([x_2+alpha x_1,…,x_n+alpha x_1])_2 is bounded by a constant?

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