Optimal exponent for arbitrary convex bodies in Dvoretzky’s theorem

Determine whether the exponent (ell+1)/2 in the dimension bound for approximately Euclidean ell-dimensional sections of arbitrary convex bodies can be improved to (ell-1)/2 when ellgreater than or equal to3.

Background

The paper defines N(ell,eps) as the smallest ambient dimension guaranteeing an approximately Euclidean ell-dimensional section for every convex body. Its main theorem gives an upper bound whose dependence on eps has exponent (ell+1)/2, while analysis of the cube yields a lower bound with exponent (ell-1)/2. The authors note that the upper-bound exponent is already suboptimal for ell=2 and leave unresolved whether the lower-bound exponent is attainable for arbitrary convex bodies when ellgreater than or equal to3.

References

We do not know whether the exponent $(\ell+1)/2$ can be improved to $(\ell-1)/2$ for arbitrary convex bodies when $\ell \geq 3$.

— A polynomial bound in Dvoretzky's theorem  (2610.03204 - Klartag et al., 2 Oct 2026) in Introduction, paragraph beginning Let us discuss the optimality of our estimates

Although we have generalized the ball construction of from the case of the cube to arbitrary convex bodies, we do not know how to generalize the sphere construction.

— A polynomial bound in Dvoretzky's theorem  (2610.03204 - Klartag et al., 2 Oct 2026) in Introduction, paragraph immediately following the discussion of the optimal exponent for the cube