Invariants in convex geometry and Banach space theory

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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

$|K|=1=|D_N|$ .

Add remark

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 $1$ ?
Add remark

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 $n$ ?
Add remark

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.