تقرير
A Pythagorean Theorem for Volume
| العنوان: | A Pythagorean Theorem for Volume |
|---|---|
| المؤلفون: | Ancel, Fredric D. |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Metric Geometry, 51N20 (Primary) 51F99 15A75 28A75 (Secondary) |
| الوصف: | Lebesgue measurable subsets A and B of parallel or identical k-dimensional affine subspaces of Euclidean n-space E^n satisfy The Product Formula for Volume: Vol_k(A)Vol_k(B) = \sum_{J \in S(n,k)} Vol_k({\pi}_J(A))Vol_k({\pi}_J(B)). Here Vol_k denotes k-dimensional Lebesgue measure; S(n,k) denotes the set of all k-element subsets of {1,2,..., n}; and for J \in S(n,k), E^J = {(x_1,x_2,...,x_n) \in E^n : x_i = 0 for all i \notin J} and {\pi}_J : E^n \rightarrow E^J is the projection that sends the i^{th} coordinate of a point of E^n to 0 whenever i \notin J. Setting B = A, we obtain the corollary: The Pythagorean Theorem for Volume: Vol_k(A)^2 = \sum_{J \in S(n,k)} (Vol_k({\pi}_J(A)))2. Comment: 19 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2305.08068 |
| رقم الأكسشن: | edsarx.2305.08068 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |