Meaning of sharp upper bound

Question

I am confused about what a "sharp" upper bound means. I have already seen this question asked elsewhere (linked) but it does not answer my question: What does it mean when a bound is sharp?

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The main question that I have is:

Is a "sharp" upper bound unique?

It is easier to work with an example, so the following is a verbose version of the question:

Consider a set of scaled sinusoidal functions, $$ S = \{f_t~|~\forall x \in \mathbb{R}~~f_t(x) = t\sin(x),~~t \in [0, 1]\}. $$ I can establish the upper bound, $$ \forall f_t \in S,~~f_t(x) \leq 1~~\forall x. $$ Clearly, the upper bound is attained for the function $f_1$. Would this upper bound be called sharp? What about the following upper bound which is clearly better? $$ \forall f_t \in S,~~f_t(x) \leq t~~\forall x. $$ Would both these bounds be called sharp?

Answer 1

I would say that a "sharp" upper bound is unique, yes. But hidden in this phrase is some subtlety: what do you mean by an "upper bound"? If we have a collection of things that we allow ourselves to call "upper bounds", we can ask whether one of those is sharp at all, but if we don't define "upper bound" clearly, it might not be obvious which objects count and which don't.

In your example, you clearly want to extend the notion of "upper bound" from a single function $f$ to a family of functions $f_t$ parametrised by some $t$. You have (at least!) two options:

• a "global" upper bound, i.e. a single upper bound for the entire family of functions, i.e. some constant $a$ such that $f_t(x) \leq a$ for all $t$ and $x$
• a "local" upper bound, i.e. a family of upper bounds for the functions, i.e. a family of constants $a_t$ such that $f_t(x) \leq a_t$ for all $t$ and $x$.

Now, all global upper bounds are local upper bounds, but not vice-versa; and something that is sharp as a global upper bound (i.e. is "best possible" among all global upper bounds) might not be sharp as a local upper bound.

• Your first bound is sharp as a global upper bound, but not sharp as a local upper bound.
• Your second bound isn't a global upper bound at all, but is a local upper bound and is sharp as such.

(I've just made up the terminology "local" and "global" here, by the way.)