Higher traces as boundary averages on finite-dimensional normed spaces

Abstract

Let $X$ be an $N$-dimensional real normed space, let $1\leqslant k\leqslant N$, and set $V=\Lambda^kX$ and $m=\binom Nk$. We characterise the probability measures $\eta$ on the unit sphere of $V$ for which \[ \operatorname{tr}(\Lambda^kA)=m\int w^\sharp\big((\Lambda^kA)w\big)d\eta(w) \] holds for every $A\in\operatorname{End}(X)$: this is equivalent to $m\int w\otimes w^\sharp d\eta(w)=\operatorname{Id}_V$. The cone probability measure always satisfies this condition, giving a canonical higher-trace formula for every norm. Normalised Euclidean hypersurface measure also does so under a scalar-commutant symmetry hypothesis, including spaces with a $1$-symmetric basis. We further obtain atomic and polyhedral formulae and show that, within a natural power-weighted family, cone measure is the unique universally isotropic member; for hypersurface measure the first-order obstruction is precisely the degree-$2$ spherical harmonic component of the support function.