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John's theorem states that to any full-dimensional symmetric convex set $K\subseteq R^n$ and any Ellipsoid $E\subseteq R^n$ that is centered at origin, there exists an invertible linear map $T$ so that $E\subseteq T(K)\subseteq\sqrt{n}E$.

Is the $\sqrt{n}$ quantity sharp or could it be improved? Is there special useful scenarios an improvement could be made?

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    $\begingroup$ The cube witnesses that it is sharp. Look at Chapter 12 in the book by Albiac and Kalton, or, for more, consult Nicole Tomczak-Jaegermann's book. $\endgroup$
    – Bill Johnson
    Commented Mar 21, 2015 at 22:30
  • $\begingroup$ @BillJohnson Thank you. Any situations we could hope to improve? $\endgroup$
    – Turbo
    Commented Mar 21, 2015 at 22:38
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    $\begingroup$ Sure, if the norm generated by the body has good properties such as type. For example, if the resulting space embeds into $L_p$ isometrically the upper estimate is $n^{|1/p - 1/2|}$ (result of D. R. Lewis). $\endgroup$
    – Bill Johnson
    Commented Mar 22, 2015 at 23:15
  • $\begingroup$ @BillJohnson Could you please refer the paper by D.R. Lewis? $\endgroup$
    – Turbo
    Commented Mar 23, 2015 at 0:44
  • $\begingroup$ Could you also elaborate to an answer? $\endgroup$
    – Turbo
    Commented Mar 23, 2015 at 0:47

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