About this space subset recording form
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${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = 1, x_iin [0,1]} = triangle_{n-1}$, $overline{v} = (x_1, cdots, x_n)$
If I will change some parameters, I can have:${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = n, x_iin [0,n]}$ -- big $n-1$-simplex, whose vertices are equal to position vectors $noverline{e_i}$.
${overline{v}inmathbb{R}^n mid x_iin [0,n]} = {0,n}^n$ -- $n$-hypercube.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = const, forall x_iin mathbb{R}^1}$ -- what the "simplex" is this?
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i leq n, x_iin [-n,n]}$ something like $n$-dim "rhombus" with the inside whose vertices are equal to position vectors $noverline{e_i}$ and $-noverline{e_i}$.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i^2 leq const^2, forall x_iin mathbb{R}^1} = D^n_c (0)$ -- $n$-dim disk with center in $0$ and radius $= constant$. But here I am using the euclidean distance and the norm of vectors, right? And if I change metric in space, this condition will not satisfy by properties of disk, yes?
What else can be built with this method of description? Do other sets (excluding item 6) depend explicitly on the space metric?
May be you can give me the name of the textbook, where I can read about this recording form of $mathbb{R}^n$ subsets?
general-topology functional-analysis metric-spaces
asked Nov 16 at 9:37
Arsenii
976
add a comment |
up vote
0
down vote
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${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = 1, x_iin [0,1]} = triangle_{n-1}$, $overline{v} = (x_1, cdots, x_n)$
If I will change some parameters, I can have:${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = n, x_iin [0,n]}$ -- big $n-1$-simplex, whose vertices are equal to position vectors $noverline{e_i}$.
${overline{v}inmathbb{R}^n mid x_iin [0,n]} = {0,n}^n$ -- $n$-hypercube.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = const, forall x_iin mathbb{R}^1}$ -- what the "simplex" is this?
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i leq n, x_iin [-n,n]}$ something like $n$-dim "rhombus" with the inside whose vertices are equal to position vectors $noverline{e_i}$ and $-noverline{e_i}$.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i^2 leq const^2, forall x_iin mathbb{R}^1} = D^n_c (0)$ -- $n$-dim disk with center in $0$ and radius $= constant$. But here I am using the euclidean distance and the norm of vectors, right? And if I change metric in space, this condition will not satisfy by properties of disk, yes?
What else can be built with this method of description? Do other sets (excluding item 6) depend explicitly on the space metric?
May be you can give me the name of the textbook, where I can read about this recording form of $mathbb{R}^n$ subsets?
general-topology functional-analysis metric-spaces
asked Nov 16 at 9:37
Arsenii
976
All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = 1, x_iin [0,1]} = triangle_{n-1}$, $overline{v} = (x_1, cdots, x_n)$
If I will change some parameters, I can have:${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = n, x_iin [0,n]}$ -- big $n-1$-simplex, whose vertices are equal to position vectors $noverline{e_i}$.
${overline{v}inmathbb{R}^n mid x_iin [0,n]} = {0,n}^n$ -- $n$-hypercube.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = const, forall x_iin mathbb{R}^1}$ -- what the "simplex" is this?
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i leq n, x_iin [-n,n]}$ something like $n$-dim "rhombus" with the inside whose vertices are equal to position vectors $noverline{e_i}$ and $-noverline{e_i}$.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i^2 leq const^2, forall x_iin mathbb{R}^1} = D^n_c (0)$ -- $n$-dim disk with center in $0$ and radius $= constant$. But here I am using the euclidean distance and the norm of vectors, right? And if I change metric in space, this condition will not satisfy by properties of disk, yes?
What else can be built with this method of description? Do other sets (excluding item 6) depend explicitly on the space metric?
May be you can give me the name of the textbook, where I can read about this recording form of $mathbb{R}^n$ subsets?
general-topology functional-analysis metric-spaces
asked Nov 16 at 9:37
Arsenii
976
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = 1, x_iin [0,1]} = triangle_{n-1}$, $overline{v} = (x_1, cdots, x_n)$
If I will change some parameters, I can have:${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = n, x_iin [0,n]}$ -- big $n-1$-simplex, whose vertices are equal to position vectors $noverline{e_i}$.
${overline{v}inmathbb{R}^n mid x_iin [0,n]} = {0,n}^n$ -- $n$-hypercube.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i = const, forall x_iin mathbb{R}^1}$ -- what the "simplex" is this?
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i leq n, x_iin [-n,n]}$ something like $n$-dim "rhombus" with the inside whose vertices are equal to position vectors $noverline{e_i}$ and $-noverline{e_i}$.
${overline{v}inmathbb{R}^n mid sum_{i = 1}^{n} x_i^2 leq const^2, forall x_iin mathbb{R}^1} = D^n_c (0)$ -- $n$-dim disk with center in $0$ and radius $= constant$. But here I am using the euclidean distance and the norm of vectors, right? And if I change metric in space, this condition will not satisfy by properties of disk, yes?
What else can be built with this method of description? Do other sets (excluding item 6) depend explicitly on the space metric?
May be you can give me the name of the textbook, where I can read about this recording form of $mathbb{R}^n$ subsets?
general-topology functional-analysis metric-spaces
general-topology functional-analysis metric-spaces
asked Nov 16 at 9:37
Arsenii
976
asked Nov 16 at 9:37
Arsenii
976
edited Nov 17 at 12:10
asked Nov 16 at 9:37
Arsenii
976
asked Nov 16 at 9:37
Arsenii
976
asked Nov 16 at 9:37
Arsenii
976
976
All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10
add a comment |
All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10
All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10
add a comment |
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All your sums must begin with $i=1$. You should also say that $bar v = (x_1,dots,x_n)$.
– Paul Frost
Nov 17 at 12:04
@PaulFrost ye, you right. I corrected it
– Arsenii
Nov 17 at 12:10