Prove that the domain of a concave function is convex. The domain of f is defined asdom f := { x ∈ R n ∣ f ( x ) &lt; + ∞ }

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Prove that the domain of a concave function is convex. The domain of f is defined asdom  f  :=  {  x  ∈            R        n    ∣  f  (  x  )  &amp;lt;  +  ∞  }

Steinherrjm · Accepted Answer

Typically a concave function is extended by defining it to be   −  ∞ outside its domain. Since we usually maximize a concave   f, &quot;infinitely bad&quot; here means we define the extension             f      ~        (  x  )  =  −  ∞ for   x  ∉  dom   f.If we define concavity, for the extended function             f      ~      , as:  ∀  x  ,  y  ,  0  &amp;lt;  θ  &amp;lt;  1  ,            f      ~        (  θ  x  +  (  1  −  θ  )  y  )  ≥  θ            f      ~        (  x  )  +  (  1  −  θ  )            f      ~        (  y  )then it&#039;s not hard to see why   dom   f  :=  {  x      |              f      ~        (  x  )  &amp;gt;  −  ∞  } is convex by our definition.

Prove that the domain of a concave function is convex. The domain of f is defined as dom f:={x in R^n∣f(x)<+oo}

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