Non-continuous valuations on convex bodies and a new characterization of volume
Jorge S. Ibáñez Marcos, Pedro Tradacete, Ignacio Villanueva
TL;DR
The paper investigates replacing continuity by boundedness in valuation theory on convex bodies, aiming to extend Hadwiger’s volume characterization. By restricting attention to parallelotopes Pa$(\{e_i\})$ and employing a grid-based counting argument, it proves that a translation-invariant, $n$-homogeneous, bounded valuation on $\mathcal{K}^n$ is necessarily a constant multiple of the Lebesgue measure, thus generalizing Hadwiger’s result under boundedness. It also identifies the admissible homogeneity degrees for bounded valuations as $[0,n-1]\cup\{n\}$ and constructs curvature-based examples $\phi_l(K)=\int_{\partial K} \kappa(K,x)^l d\mathcal{H}^{n-1}(x)$ to realize all degrees in $[0,n-1]$, underscoring the sharpness of the counting approach. The work further highlights the limitations of automatic continuity beyond restricted classes by presenting counterexamples to McMullen-type integral representations in the absence of continuity. Overall, the results illuminate how boundedness can substitute continuity in certain Hadwiger-type characterizations while clarifying the boundaries of this method in general valuation theory.
Abstract
This paper investigates the use of automatic continuity techniques in the context of valuations on convex bodies. We first provide an automatic continuity theorem for valuations restricted to parallelotopes with respect to a fixed basis. This result in combination with a counting argument provides a strengthened version of a classical characterization of volume due to Hadwiger. As a byproduct of the proof it is shown that $[0,n-1]\cup\{n\}$ are precisely the possible degrees of homogeneity of bounded translation invariant valuations on $n$-dimensional convex bodies.