Is this function convex or concave on ( x , y , z )? A, B, a, b, and c are positive constants. f ( x , y , z ) = A exp ⁡ ( ( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 ) + B exp ⁡ ( x 2 + y 2 + z 2 )

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Is this function convex or concave on   (  x  ,  y  ,  z  )?   A,   B,   a,   b, and   c are positive constants.  f  (  x  ,  y  ,  z  )  =  A  exp  ⁡      (          (      x      −      a              )        2            +      (      y      −      b              )        2            +      (      z      −      c              )        2              )    +  B  exp  ⁡      (                  x        2            +              y        2            +              z        2              )

hitturn35 · Accepted Answer

The Euclidean norm   x  ↦      |    x      |   (and any norm for that matter) is a convex function. This follows from the definition of a norm.If   f  :            R        n    →      R   is convex and   ϕ  :      R    →      R   is a convex nondecreasing function, then the composition   ϕ  ∘  f is also convex. This is immediate from the definition of convexity used twice.The sum of two convex functions is convex.

Is this function convex or concave on (x,y,z)? A, B, a, b, and c are positive constants. f(x,y,z)=Aexp( sqrt((x−a)^2+(y−b)^2+(z−c)^2)+Bexp(sqrt (x^2+y^2+z^2)

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