Sharp projection-body inequalities and a counterexample to simplex maximization

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

• Petty’s projection-volume conjecture in dimensions at least four
• A product counterexample to the simplex maximum for projection-body volume

Scope

Petty's projection-volume conjecture predicts

$\displaystyle \frac{|\Pi K|}{|K|^{n-1}}\ge \kappa_{n-1}^{n}\kappa_n^{2-n},$

with equality exactly for ellipsoids; here $\kappa_j$ is the volume of the Euclidean unit ball in dimension $j$. The formalization establishes this for every convex body $K\subset\mathbb R^n$ and every $n\ge4$.

Other inequalities in the projection-body family are not included.

Brannen's simplex-maximization conjecture predicts that a simplex maximizes normalized projection-body volume $|\Pi K|/|K|^{n-1}$ in dimension $n$. The formalized counterexample is the product of two ten-dimensional simplices. Its normalized projection volume, divided by that of a twenty-dimensional simplex, is

$\displaystyle \frac{22{,}355{,}476}{22{,}020{,}096}>1.$

Thus the proposed maximum fails in dimension $20$. The paper's exponential-factor result for every sufficiently large dimension is not included.

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