Minkowski functionals (valuations)

Question

What can we say about a polyconvex set $\omega\subset\mathbb{R}^N$ knowing the values of its Minkowski valuations?

I know that Hadwiger characterization theorem states that there are only $N+1$ valuations that can express any continuous, rigid motion invariant valuation as a linear combination. But what does that say about $\omega$. Can we determine something about its shape? Or can we characterize the set of all polyconvex sets of $\mathbb{R}^N$ that have the same values of the $N+1$ Minkowski functionals?

For details you can see Daniel Klain & Gian-Carlo Rota - Introduction Geometric Probability (page 118 - Hadwiger's Theorem).

Answer 1

I doubt that much can be said about the shape of a convex body given its intrinsic volumes (this is a more common name for the basic valuations). Intuitively, this is because the space of convex bodies is "infinite dimensional".

Assume that a body $K$ has a hyperplane section with symmetries (for example, this section is a simplex or a ball). If we cut $K$ along this section and glue the pieces back by applying a symmetry of the section, then the result has the same intrinsic volumes by the valuation formula.

For many convex bodies there is a parallelopiped with the same values of intrinsic volumes.

One can ask a related question: given some special family of convex bodies in $\mathbb{R}^d$ depending on $d$ parameters, can these bodies be distinguished by the values of their intrinsic volumes? The answer is yes for parallelopipeds in all dimensions and for ellipsoids in $\mathbb{R}^3$, see https://arxiv.org/abs/1905.01728. However, the answer seems to be unknown for ellipsoids of higher dimension.

Finally, in some extreme cases the intrinsic volumes determine the body uniquely: for example if the isoperimetric inequality holds with equality, then the body is a ball.

Answer 2

The kinematic formula for example says you can retrieve the expected Euler characteristic of the set intersected with a randomly translated and rotated shape, whatever that shape is. The Minkowski valuations contain in fact exactly this information, not more.