球面与双曲凸集上的单调哈德维格定理
Monotone Hadwiger Theorems on Spherical and Hyperbolic Convex Sets
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中文总结 AI 辅助
该研究分类了球面与双曲凸集上单调的等距不变估值，无需连续性假设，给出唯一表示及系数条件，并证明单调性蕴含连续性与对称性。
中文摘要 AI 辅助
我们分类了在非空球面凸集上单调的实值$\SO(n+1)$-不变估值，无需假设连续性或可测性。在真闭球面凸集上（并加入空集），非空集合间的单调性等价于在归一化球面均质积分中的唯一表示$\sum_{j=0}^{n}c_jU_j$，其中$c_0\in\R$且$c_1,\ldots,c_n\geq0$；包含空集的单调性等价于$c_0\geq0$。在所有闭球面凸集上，非空集合间的单调性等价于在球面内蕴体积中的唯一展开$\sum_{j=0}^n a_jv_j^{\mathrm s}$，满足$0\leq a_0\leq\cdots\leq a_n$，并且已经蕴含包含空集的单调性。在这两种球面情形中，单调性蕴含连续性和$\Orth(n+1)$-不变性。对于每个$n\geq1$，紧致双曲凸集上的连续等距不变估值构成$\chi,W_0^{(n)},\ldots,W_{n-1}^{(n)}$的线性张成。非空集合间的单调性等价于非负的均质积分系数，欧拉系数任意，并且同样蕴含连续性。真球面分类还给出了尖闭凸锥的等价表述，并下推到实椭圆空间中的紧致射影凸集，从而得到相应的连续和单调分类。
英文摘要
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.
