[幾何] R^n中Ball位移問題
符號：1.B_p(r)：以p為圓心，r為半徑的compact ball
2.o=(0,0,0,0,...,0)
3.r-ball：半徑為a的compact ball
問題：將B_o(r)往下位移至B_p(r) , p = (0,0,...,-a), 0<a<r
(即往下位移a單位，但不超過r)
試證：B_o(r)的下半球會被
"B_p(r)的上半球" 聯集 "B_o(r)赤道球殼每點所形成的a-ball聯集" 給蓋住
附個R^2的圖解釋：http://imgur.com/s66DieD
(藍色與紅色加起來是上球的下半球，而紅色部分可以由下球蓋住
但藍色部分無法，需要箭頭那兩個點去形成a-ball才能蓋住)
但畢竟最高只能到R^3去想像(赤道那圈去形成a-ball)，更高維度就無法了
翻譯成數學的語言的話，即是
Let B_o(r), B_p(r) be two compact balls defined as above
where p=(0,0,...,-a) , 0<a<r
Define S = {(x_1,x_2,...,x_(n-1),0)│Σ(x_i)^2 = r^2}
U = {(x_1,,....,x_(n-1),x_n)│Σ(x_i)^2 = r^2, x_n≦0}
Prove U is included in B_p(r) ∪ ( ∪ B_x(a) )
x€S
(S就是原球的赤道，U就是原球的下半球，B_p(r)是下移後的球，∪B_x(a)是赤道每點
張出的a-ball聯集 )
謝謝！
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※ 編輯: znmkhxrw (61.231.125.61), 03/27/2015 20:15:03
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