Is Hlawkas Inequality holds for sobolev space

Question

im wondring is that inequality holds for any functionnal space such as sobolev space

and if it's true how we can write it in that space

/HlawkasInequality

any help would be apperciated

Answer 1

This is a long comment. Quote from Hlawka's functional inequality:

Moreover, Witsenhausen showed that the space $L^p(0, 1)$ is a Hlawka space for $1\le p\le 2$. Therefore, one can see that all Banach spaces having the property that all its finite dimensional subspaces can be embedded linearly and isometrically in the space $L^p([0, 1])$, with some $1\le p\le 2$ are Hlawka spaces (see Niculescu and Persson and Lindenstrauss and Pelczy'nski). Further, Witsenhausen proved that a finite-dimensional real space with piecewise linear norm is embeddable in $L^1$ if and only if it is a Hlawka space. However, Neyman showed that in the general case embeddability in $L^1$ does not characterize Hlawka spaces. Concluding, to the best of the author’s knowledge, no characterization of Hlawka spaces is presently known.