Let h 1 and h 2 be two differentiable convex functions on R n and x ¯ ∈ R n such that h 1 ( x ¯ ) = h 2 ( x ¯ ) = 0 and ∇ h 1 ( x ¯ ) = λ ∇ h 2 ( x ¯ ) for some λ &gt; 0 . Prove that there exists a neighborhood U of x ¯ such that one of the following two sets contains the other one: { x ∈ U : h 1 ( x ) ≥ 0 } and { x ∈ U : h 2 ( x ) ≥ 0 } .

Question

Let       h    1   and       h    2   be two differentiable convex functions on             R        n   and             x      ¯        ∈            R        n   such that       h    1    (            x      ¯        )  =      h    2    (            x      ¯        )  =  0 and   ∇      h    1    (            x      ¯        )  =  λ  ∇      h    2    (            x      ¯        ) for some   λ  &amp;gt;  0 . Prove that there exists a neighborhood U of             x      ¯       such that one of the following two sets contains the other one:   {  x  ∈  U  :      h    1    (  x  )  ≥  0  } and   {  x  ∈  U  :      h    2    (  x  )  ≥  0  } .

coccusk7 · Accepted Answer

Step 1Here is a counterexample:  n  =  3  ,              x      ¯        =  (  1  ,  0  ,  0  )  ,        h    1    (  x  ,  y  ,  z  )  =      x    2    +      y    2    +      z    2    −  1  ,        h    2    (  x  ,  y  ,  z  )  =      x    2    +      1    2        y    2    +  2      z    2    −  1.In fact in small neighborhoods of             x      ¯       in xy plane   {  x  :      h    2    (  x  )  ≥  0  } is strictly smaller than   {  x  :      h    1    (  x  )  ≥  0  } (because for   y  ≠  0 ,       h    1    (  x  ,  y  ,  0  )  &amp;gt;      h    2    (  x  ,  y  ,  0  ) ) and in small neighborhoods in xz plane   {  x  :      h    2    (  x  )  ≥  0  } is strictly bigger.

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