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on Mahler volume product constant ($C_{25}$)

The statement

Let $K\subset\mathbb R^n$ be a centrally symmetric convex body (compact, convex, with non-empty interior) satisfying $K=-K$ . Its polar body is

K^\circ := \left\{y\in\mathbb R^n:\ \langle x,y\rangle \le 1\ \text{for all }x\in K\right\}.

The volume product of $K$ is

\mathrm{vp}(K) := \mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ).

It is common (and convenient) to absorb the factorial and define the Mahler volume

M(K) := n!\,\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)=n!\,\mathrm{vp}(K).

For centrally symmetric $K$ , this quantity is invariant under invertible linear transformations, so it makes sense to ask for a lower bound of the form $M(K)\ge c^n$ .

The constant $C_{25}$ is the largest $c$ such that

n!\,\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)\ \ge\ c^n \quad\text{for all }n\ge 1\text{ and all centrally symmetric convex bodies }K\subset\mathbb R^n,

equivalently

\mathrm{Vol}_n(K)\,\mathrm{Vol}_n(K^\circ)\ \ge\ \frac{c^n}{n!}.

The (symmetric) Mahler conjecture predicts that $C_{25}=4$ , with extremisers given by Hanner polytopes (in particular, the cube and cross-polytope).

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.