Seyhmus Güngören
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 Dec 20 comment on existence of supremum/infimum $X$ is not compact. what I can do is to increase the radius of the closed ball. Then it will include many other density functions. Will this help me to achive the supremum of $T$, where $T$ is a convex functional of $g$? if not what can I do? in real numbers I extend the domain(exampe 2 in the question) and I am done. Dec 20 comment on existence of supremum/infimum i do not ask the supremum of a set. I ask the supremum of a functional defined over this set. Analogy is the function given which was defined over real numbers. Now instead of real numbers as the domain, we have the set of density functions. Let $T$ a convex functional of $g$. Dec 20 comment on existence of supremum/infimum I dont get what you mean. If the space of functions are given by $X$ as in the question and if we are interested in the existence of a density function $g^*$ (f is known and given) on $X$ such that the supremum of $T$ is achieved for this $g^*$. Will I be able to guarantee the existence of such $g^*$ all the time (for any $\delta$ or even $T$). I think it is not possible. Then where should I look for such a $g^*$. I understood that supremum is more related to the completeness than it is related to the compactness. But how does it help me here? Dec 20 comment on existence of supremum/infimum I added some more to the question reagarding your answers first paragaph. Dec 20 comment on existence of supremum/infimum @coffeemath yes I meant that..thx for the information. Dec 20 revised on existence of supremum/infimum added 230 characters in body Dec 20 comment on existence of supremum/infimum @coffeemath what does $a$ have something to do with example 2? Dec 20 revised on existence of supremum/infimum deleted 32 characters in body Dec 20 revised on existence of supremum/infimum deleted 32 characters in body Dec 20 comment on existence of supremum/infimum @coffeemath sorry that was about the argument. Dec 20 asked on existence of supremum/infimum Dec 20 accepted Compactness in minimax theorem Dec 17 revised compare sample mean and sample median as estimators of µ added 217 characters in body Dec 17 comment compare sample mean and sample median as estimators of µ @Batman you are right and I know all that stuff. But when you read the question do you think robustness is a part of the question? Dec 17 answered compare sample mean and sample median as estimators of µ Nov 29 comment Comparison of constrained optimization methods Then you are looking for a person who has at least read the paper and probably also implemented? Nov 28 comment Comparison of constrained optimization methods did you implement both of them? Nov 27 comment Prove that if $f(x)$ is convex and nonnegative, then $g(x)=(f(x))^2$ is convex too I think the question is now abit different than it was before.. Nov 27 comment Random Walk And Stochastic-Processes @JoyGardenia dont thank. If you like the answer just upvote. Nov 27 answered Complementary slackness with Lagrange Multipliers in Convex Optimization