The Poincaré-constant of a domain is the smallest constant such that the estimate

holds (where is the mean value of ).

These constants are known for some classes of domains and some values of : E.g. Payne and Weinberger showed in 1960 that for and convex the constant is and Acosta and Duran showed in 2004 that for and convex one gets .

I do not know of any other results on these constants, and while playing around with these kind of things, I derived another one: Since I often work with images I will assume that in the following.

Using the co-area formula we can express the -norm of the gradient via the length of the level sets: Denoting by the -dimensional Hausdorff measure (which is, roughly speaking, the length in the case of ) and with the level set of at level , the co-area formula states that

Now we combine this with the isoperimetric inequality, which is

where is unit ball and denotes the Lebesgue measure. Combining~(1) and~(2) we get

Now we use the trivial fact that

combined with Fubini to get

Since and it holds that . Combining this with~(4) we get

Finally, we use~(3) to obtain

in other words

This estimate is quite explicit, does not need the subtraction of the mean value, does not need convexity of , but also does not obey the scaling (which is of no surprise since we used the condition which also does not obey this scaling).

In dimension the estimate takes the simpler form

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