Dual mean Minkowski measures and the Grünbaum conjecture for affine diameters

For a convex body K in a Euclidean vector space χ of dimension n (≥ 2), we define two subarithmetic monotonic sequences (σK,k)_k≥1 and (σ_K^o,K,k)_k≥1 of functions on the interior of K. The k-th members are "mean Minkowski measures in dimension k" which are pointwise dual: σ_K^o(z) = σK^z,k(z), where z ∈ int K, and K^z is the dual (polar) of K with respect to z. They are measures of (anti-)symmetry of K in the following sense: 1 ≤ σ_K,k(z), σ_K,k^o(z) ≤ k+1/2. The lower bound is attained if and only if K has a k-dimensional simplicial slice or simplicial projection. The upper bound is attained if and only if K is symmetric with respect to z. In 1953 Klee showed that the lower bound m* _K > n - 1 on the Minkowski measure of K implies that there are n C 1 affine diameters meeting at a critical point z* ∈ K. In 1963 Grünbaum conjectured the existence of such a point in the interior of any convex body (without any conditions). While this conjecture remains open (and difficult), as a byproduct of our study of the dual mean Minkowski measures, we show that n / m*_K+1 ≤ σ_K,n-1^o(z*) always holds, and for sharp inequality Grünbaum's conjecture is valid.