{"id":947,"date":"2014-05-31T20:59:11","date_gmt":"2014-06-01T00:59:11","guid":{"rendered":"http:\/\/www.curtisbright.com\/bln\/?p=947"},"modified":"2014-05-31T20:59:11","modified_gmt":"2014-06-01T00:59:11","slug":"volume-of-a-hypersphere-in-the-1-norm","status":"publish","type":"post","link":"http:\/\/localhost\/blog\/index.php\/2014\/05\/31\/volume-of-a-hypersphere-in-the-1-norm\/","title":{"rendered":"Volume of a hypersphere in the 1-norm"},"content":{"rendered":"

Previously<\/a>\u00a0I derived the volume of a hypersphere in $n$ dimensions. A hypersphere with radius $R$ consists of the set of points $\\newcommand{\\x}{\\mathbf{x}}\\newcommand{\\R}{\\mathbb{R}}\\x=(x_1,\\dotsc,x_n)\\in\\R^n$ for which<\/p>\n

\\[ \\lVert\\x\\rVert \\leq R , \\]<\/p>\n

where $\\lVert\\x\\rVert$ denotes the usual Euclidean norm<\/a> (also known as the 2-norm),<\/p>\n

\\[ \\lVert\\x\\rVert := \\sqrt{x_1^2+\\dotsb+x_n^2} .\u00a0\\]<\/p>\n

Today, I’d like to consider the problem of computing the volume of an $n$-dimensional hyphersphere in the 1-norm<\/a> (also known as the Manhattan distance or taxicab norm), which is defined by<\/p>\n

\\[ \\lVert\\x\\rVert_1 := \\lvert x_1\\rvert+\\dotsb+\\lvert x_n\\rvert .\u00a0\\]<\/p>\n

The volume of the 1-norm hypersphere is given by the expression<\/p>\n

\\[ V_n(R) := \\frac{(2R)^n}{n!} , \\]<\/p>\n

as we will show by induction on $n$. In the base case $n=1$ one has<\/p>\n

\\[ \\newcommand{\\d}{\\,\\mathrm{d}} V_1(R) = \\int_{-R}^R\\d x_1 = 2R , \\]<\/p>\n

as required. Now suppose that the formula holds in dimension $n-1$. Then we have<\/p>\n

\\begin{align*}
\nV_n(R) &= \\int\\limits_{\\lvert x_1\\rvert\\leq R}\\;\\int\\limits_{\\lvert x_1\\rvert+\\lvert x_2\\rvert\\leq R}\\dotsi\\int\\limits_{\\lvert x_1\\rvert+\\dotsb+\\lvert x_n\\rvert\\leq R}\\d x_n\\dotsm\\d x_1 \\\\
\n&=\u00a0\\int\\limits_{\\lvert x_1\\rvert\\leq R}\\;\\int\\limits_{\\lvert x_2\\rvert\\leq R-\\lvert x_1\\rvert}\\dotsi\\int\\limits_{\\lvert x_2\\rvert\\dotsb+\\lvert x_n\\rvert\\leq R-\\lvert x_1\\rvert}\\d x_n\\dotsm\\d x_1 \\\\
\n&=\u00a0\\int\\limits_{\\lvert x_1\\rvert\\leq R} V_{n-1}\\bigl(R-\\lvert x_1\\rvert\\bigr) \\d x_1 \\\\
\n&= \\int_{-R}^R \\frac{2^{n-1}(R-\\lvert x_1\\rvert)^{n-1}}{(n-1)!} \\d x_1 \\\\
\n&= 2\\int_{0}^R \\frac{2^{n-1}(R-x_1)^{n-1}}{(n-1)!} \\d x_1 \\\\
\n&= \\frac{2^n}{(n-1)!}\\biggl[-\\frac{1}{n}(R-x_1)^n\\biggr]_0^R \\\\
\n&= \\frac{(2R)^n}{n!}
\n\\end{align*}<\/p>\n

By induction, the formula holds for all positive integers $n$.<\/p>\n","protected":false},"excerpt":{"rendered":"

Previously\u00a0I derived the volume of a hypersphere in $n$ dimensions. A hypersphere with radius $R$ consists of the set of points $\\newcommand{\\x}{\\mathbf{x}}\\newcommand{\\R}{\\mathbb{R}}\\x=(x_1,\\dotsc,x_n)\\in\\R^n$ for which \\[ \\lVert\\x\\rVert \\leq R , \\] where $\\lVert\\x\\rVert$ denotes the usual Euclidean norm (also known as … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/947"}],"collection":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=947"}],"version-history":[{"count":0,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/posts\/947\/revisions"}],"wp:attachment":[{"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=947"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}