Well I managed to prove the symmetry.
d(x,y) = sqrt ( (x1 - y1)^2 + (x2-y2)^2 ) for all x,y in R^2
x= (x1,y1) y=(x2,y2)
Also clear is d(x,y)=0 x=y. The most complex part is the so-called triangle inequality which in the plane R^2 simply says that in a triangle the sum of two sides is not smaller than the third side of the triangle. The proof is usually done by proving the Cauchy-Schwartz inequality as an intermediate step (which is important also by itself). You should follow this path in any book or Wikipedia. It can be considered an indispensible part of any mathematical education to see in some cases geometrical evidence and algebraical proofs at work at the same theme.
function f smooth cost function over compact convex domain, Omega belong to R2, minf(x) and maxf(x) attain at a single point, so that maxf(x)>f(x)>minf(x) for all x. is it possible that all the level sets determined by y belong to max f and min f , are not convex? any specific example?
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