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1. Between Santalo and Petty Projection Inequality

Let $K$ be a convex body in $\R^N$ and $1\leq n \leq N-1$. Define $$ \Phi_{[n]}(K) := \left( \int_{G_{N,n}} |P_F(K)|^{-N} d\mu(F) \right)^{-\frac{1}{nN}}. $$ Assume in the following that $|K|=1=|D_N|$.
    1. Problem 1.1.

      If $K$ has barycenter at $0$, is it true that $\Phi_{[n]}(K) \simeq \Phi_{[n]}(D_N)$? The lower bound is true but is it true with constant $1$?
        • Problem 1.2.

          Is it true that $\Phi_{[1]}(K) \geq \Phi_{[1]}(D_N)$ for every compact $K$? Also for $N-1$ or any $n$?
            • Problem 1.3.

              Is there an argument or proof that gives Santalo and Petty projection inequality at the same time?

                  Cite this as: AimPL: Invariants in convex geometry and Banach space theory , available at http://aimpl.org/convexbanach.