Monotone Hadwiger Theorems on Spherical and Hyperbolic Convex Sets

Abstract

For every $n\geq1$, we classify monotone rotation-invariant real-valued valuations on closed spherical convex sets, without assuming continuity or measurability. On proper sets, namely those contained in an open hemisphere, these are precisely the nonnegative linear combinations of the normalized spherical quermassintegrals. On all closed spherical convex sets, they are precisely the linear combinations of the spherical intrinsic volumes with nonnegative, nondecreasing coefficients. The representations are unique, and all such valuations are continuous and invariant under the full orthogonal group. In hyperbolic space, an isometry-invariant real-valued valuation on compact convex sets is continuous if and only if it is a linear combination of the Euler characteristic and the hyperbolic quermassintegrals. This representation is unique. Monotonicity is equivalent to nonnegative coefficients and implies continuity. If monotonicity is required only between nonempty sets, the Euler coefficient is unrestricted in the proper spherical and hyperbolic cases, whereas the classification on all closed spherical convex sets is unchanged. We also obtain corresponding classifications for valuations on closed convex cones that vanish at the zero cone and monotone classifications on compact projectively convex sets contained in an affine chart of real elliptic space.