Monotone Hadwiger Theorems on Spherical and Hyperbolic Convex Sets

Abstract

We classify real-valued, $\SO(n+1)$-invariant valuations that are monotone among nonempty spherical convex sets, without assuming continuity or measurability. On proper closed spherical convex sets, with the empty set adjoined, monotonicity among nonempty sets is equivalent to a unique representation ${\sum }_{j=0}^n{c}_j{U}_j$ in the normalized spherical quermassintegrals, where $c_0\in \mathbb{R}$ and $c_1,\dots ,c_n\ge 0$; monotonicity including the empty set is equivalent to $c_0\ge 0$. On all closed spherical convex sets, monotonicity among nonempty sets is equivalent to a unique expansion ${\sum }_{j=0}^n{a}_j{v}_j^{\mathrm{s}}$ in the spherical intrinsic volumes with $0\le a_0\le \cdots \le a_n$, and already implies monotonicity including the empty set. In both spherical settings, monotonicity implies continuity and $\Orth(n+1)$-invariance. For every $n\ge 1$, the continuous isometry-invariant valuations on compact hyperbolic convex sets form the span of $\chi ,W_0^{(n)},\dots ,W_{n-1}^{(n)}$. Monotonicity among nonempty sets is equivalent to nonnegative quermassintegral coefficients, with an arbitrary Euler coefficient, and again implies continuity. The proper spherical classifications also yield an equivalent formulation for pointed closed convex cones and descend to compact projectively convex sets in real elliptic space, giving the corresponding continuous and monotone classifications.

30 pages