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on Reverse Brunn-Minkowski constant ($C_{70}$)

The statement

For subsets $K,L\subset\mathbb{R}^n$ , their Minkowski sum is

K+L\ :=\ \{x+y:\ x\in K,\ y\in L\}.

In general, one cannot expect a reverse Brunn-Minkowski inequality for arbitrary compact sets, even with a fixed multiplicative constant. [Bas1995-no-general-reverse]

Milman's reverse Brunn-Minkowski theorem says that for centrally symmetric convex bodies, after putting the bodies in a suitable relative position, one does get a dimension-free reverse inequality. [Bas1995-milman-statement]

More precisely, define $C_{\mathrm{RBM}}$ as the infimum of constants $C>0$ such that for every $n\ge 1$ and every pair of centrally symmetric convex bodies $B_1,B_2\subset\mathbb{R}^n$ , there exists a linear map $u$ with $\lvert\det u\rvert=1$ satisfying

\lvert u(B_1)+B_2\rvert^{1/n}\ \le\ C\bigl(\lvert B_1\rvert^{1/n}+\lvert B_2\rvert^{1/n}\bigr).

[Bas1995-milman-statement]

We define

C_{70}\ :=\ C_{\mathrm{RBM}},

the reverse Brunn-Minkowski constant.

Milman's theorem implies $C_{\mathrm{RBM}}<\infty$ . [Bas1995-milman-statement]

A trivial lower bound is $C_{\mathrm{RBM}}\ge 1$ .

Thus the best established range currently is

1\ \le\ C_{70}\ <\ \infty.

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.