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- W3089184329 abstract "Let $({mathbf X},|cdot|_{mathbf X}), ({mathbf Y},|cdot|_{mathbf Y})$ be normed spaces with ${mathrm{dim}}({mathbf X})=n$. Bourgain's almost extension theorem asserts that for any ${varepsilon}>0$, if ${mathcal{N}}$ is an ${varepsilon}$-net of the unit sphere of ${mathbf X}$ and $f:{mathcal{N}}to {mathbf Y}$ is $1$-Lipschitz, then there exists an $O(1)$-Lipschitz $F:{mathbf X}to {mathbf Y}$ such that $|F(a)-f(a)|_{mathbf Y}lesssim n{varepsilon}$ for all $ain mathcal{N}$. We prove that this is optimal up to lower order factors, i.e., sometimes $max_{ain {mathcal{N}}} |F(a)-f(a)|_{mathbf Y}gtrsim n^{1-o(1)}{varepsilon}$ for every $O(1)$-Lipschitz $F:{mathbf X}to {mathbf Y}$. This improves Bourgain's lower bound of $max_{ain {mathcal{N}}} |F(a)-f(a)|_{mathbf Y}gtrsim n^{c}{varepsilon}$ for some $0<c<frac12$. If ${mathbf X}=ell_2^n$, then the approximation in the almost extension theorem can be improved to $max_{ain {mathcal{N}}} |F(a)-f(a)|_{mathbf Y}lesssim sqrt{n}{varepsilon}$. We prove that this is sharp, i.e., sometimes $max_{ain {mathcal{N}}} |F(a)-f(a)|_{mathbf Y}gtrsim sqrt{n}{varepsilon}$ for every $O(1)$-Lipschitz $F:ell_2^nto {mathbf Y}$." @default.
- W3089184329 created "2020-10-01" @default.
- W3089184329 creator A5052985577 @default.
- W3089184329 date "2020-09-23" @default.
- W3089184329 modified "2023-09-27" @default.
- W3089184329 title "Impossibility of almost extension" @default.
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