Exterior Covering Number of $epsilon /2$ is greater of equal than Covering Number of $epsilon$
Definition $(epsilon -Net)$ :let $(T,d)$ be a metric space .Consider a subset $ K subset T$ and let $epsilon >0$, A subset $N subset K $ is called $epsilon -Net$ of $K$ if every point in $K$ is within a distance $epsilon$ if every point in K is within a distance $epsilon$ of some point of $N$,i.e.
$$forall x in K exists x_0 in N : d (x,x_0) leq epsilon$$
Definition(Covering Number): For metric space $(T,d)$ The covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N(K,d,epsilon )$, is the smallest possible cardinarity an $epsilon -Net$ of K , or equivalently ,is the smallest number of closed balls with centers in $K$ and radii $epsilon$ whose union covers $K$
Definition(Exterior Covering Number): For metric space $(T,d)$ The exterior covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N^{ext}(K,d,epsilon )$, is the smallest number of closed balls with centers not necessary in $K$ and radii $epsilon$ whose union covers $K$
then I was asked to prove that:
$$N(K,d,epsilon) leq N^{ext} (K,d,epsilon /2) $$
how to see that ?
here is my attempt: since each $epsilon -ball$ in $N(K,d, epsilon) $ should intersect at least one $epsilon /2 -ball$ in $N^{ext}(N,d,epsilon/2)$ ,thus I am trying to show that by contradiction: for each two distinct $epsilon -ball$ in $N(K,d, epsilon)$ ,the $epsilon /2 -balls$ they intersect must contain a distinct one.
real-analysis probability general-topology functional-analysis
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
add a comment |
Definition $(epsilon -Net)$ :let $(T,d)$ be a metric space .Consider a subset $ K subset T$ and let $epsilon >0$, A subset $N subset K $ is called $epsilon -Net$ of $K$ if every point in $K$ is within a distance $epsilon$ if every point in K is within a distance $epsilon$ of some point of $N$,i.e.
$$forall x in K exists x_0 in N : d (x,x_0) leq epsilon$$
Definition(Covering Number): For metric space $(T,d)$ The covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N(K,d,epsilon )$, is the smallest possible cardinarity an $epsilon -Net$ of K , or equivalently ,is the smallest number of closed balls with centers in $K$ and radii $epsilon$ whose union covers $K$
Definition(Exterior Covering Number): For metric space $(T,d)$ The exterior covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N^{ext}(K,d,epsilon )$, is the smallest number of closed balls with centers not necessary in $K$ and radii $epsilon$ whose union covers $K$
then I was asked to prove that:
$$N(K,d,epsilon) leq N^{ext} (K,d,epsilon /2) $$
how to see that ?
here is my attempt: since each $epsilon -ball$ in $N(K,d, epsilon) $ should intersect at least one $epsilon /2 -ball$ in $N^{ext}(N,d,epsilon/2)$ ,thus I am trying to show that by contradiction: for each two distinct $epsilon -ball$ in $N(K,d, epsilon)$ ,the $epsilon /2 -balls$ they intersect must contain a distinct one.
real-analysis probability general-topology functional-analysis
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
add a comment |
Definition $(epsilon -Net)$ :let $(T,d)$ be a metric space .Consider a subset $ K subset T$ and let $epsilon >0$, A subset $N subset K $ is called $epsilon -Net$ of $K$ if every point in $K$ is within a distance $epsilon$ if every point in K is within a distance $epsilon$ of some point of $N$,i.e.
$$forall x in K exists x_0 in N : d (x,x_0) leq epsilon$$
Definition(Covering Number): For metric space $(T,d)$ The covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N(K,d,epsilon )$, is the smallest possible cardinarity an $epsilon -Net$ of K , or equivalently ,is the smallest number of closed balls with centers in $K$ and radii $epsilon$ whose union covers $K$
Definition(Exterior Covering Number): For metric space $(T,d)$ The exterior covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N^{ext}(K,d,epsilon )$, is the smallest number of closed balls with centers not necessary in $K$ and radii $epsilon$ whose union covers $K$
then I was asked to prove that:
$$N(K,d,epsilon) leq N^{ext} (K,d,epsilon /2) $$
how to see that ?
here is my attempt: since each $epsilon -ball$ in $N(K,d, epsilon) $ should intersect at least one $epsilon /2 -ball$ in $N^{ext}(N,d,epsilon/2)$ ,thus I am trying to show that by contradiction: for each two distinct $epsilon -ball$ in $N(K,d, epsilon)$ ,the $epsilon /2 -balls$ they intersect must contain a distinct one.
real-analysis probability general-topology functional-analysis
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
Definition $(epsilon -Net)$ :let $(T,d)$ be a metric space .Consider a subset $ K subset T$ and let $epsilon >0$, A subset $N subset K $ is called $epsilon -Net$ of $K$ if every point in $K$ is within a distance $epsilon$ if every point in K is within a distance $epsilon$ of some point of $N$,i.e.
$$forall x in K exists x_0 in N : d (x,x_0) leq epsilon$$
Definition(Covering Number): For metric space $(T,d)$ The covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N(K,d,epsilon )$, is the smallest possible cardinarity an $epsilon -Net$ of K , or equivalently ,is the smallest number of closed balls with centers in $K$ and radii $epsilon$ whose union covers $K$
Definition(Exterior Covering Number): For metric space $(T,d)$ The exterior covering number of $K subset T$ respect to a given $epsilon geq 0$ ,denotes as $N^{ext}(K,d,epsilon )$, is the smallest number of closed balls with centers not necessary in $K$ and radii $epsilon$ whose union covers $K$
then I was asked to prove that:
$$N(K,d,epsilon) leq N^{ext} (K,d,epsilon /2) $$
how to see that ?
here is my attempt: since each $epsilon -ball$ in $N(K,d, epsilon) $ should intersect at least one $epsilon /2 -ball$ in $N^{ext}(N,d,epsilon/2)$ ,thus I am trying to show that by contradiction: for each two distinct $epsilon -ball$ in $N(K,d, epsilon)$ ,the $epsilon /2 -balls$ they intersect must contain a distinct one.
real-analysis probability general-topology functional-analysis
real-analysis probability general-topology functional-analysis
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
edited Dec 28 '18 at 0:51
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
asked Dec 28 '18 at 0:43
ShaoyuPei
1698
1698
add a comment |
add a comment |
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Suppose that $$bar B(x_1, epsilon / 2), dots, bar B(x_{N^{rm ext}}, epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i in { 1, dots, N^{rm ext}}$, there exists a $k_i in K$ that is contained in $bar B(x_i, epsilon / 2)$. (Otherwise $bar B(x_i, epsilon / 2)$ would be redundant, contradicting the minimality of size of the covering).
By the triangle inequality,
$$ bar B(x_i , epsilon / 2) subseteq bar B(k_i, epsilon), $$
for each $i$, which means that
$$bar B(k_1, epsilon ), dots, bar B(k_{N^{rm ext}}, epsilon)$$
is an internal covering of size $N^{rm ext}$. Hence the smallest internal covering has size at most $N^{rm ext}$.
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
add a comment |
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Suppose that $$bar B(x_1, epsilon / 2), dots, bar B(x_{N^{rm ext}}, epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i in { 1, dots, N^{rm ext}}$, there exists a $k_i in K$ that is contained in $bar B(x_i, epsilon / 2)$. (Otherwise $bar B(x_i, epsilon / 2)$ would be redundant, contradicting the minimality of size of the covering).
By the triangle inequality,
$$ bar B(x_i , epsilon / 2) subseteq bar B(k_i, epsilon), $$
for each $i$, which means that
$$bar B(k_1, epsilon ), dots, bar B(k_{N^{rm ext}}, epsilon)$$
is an internal covering of size $N^{rm ext}$. Hence the smallest internal covering has size at most $N^{rm ext}$.
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
add a comment |
Suppose that $$bar B(x_1, epsilon / 2), dots, bar B(x_{N^{rm ext}}, epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i in { 1, dots, N^{rm ext}}$, there exists a $k_i in K$ that is contained in $bar B(x_i, epsilon / 2)$. (Otherwise $bar B(x_i, epsilon / 2)$ would be redundant, contradicting the minimality of size of the covering).
By the triangle inequality,
$$ bar B(x_i , epsilon / 2) subseteq bar B(k_i, epsilon), $$
for each $i$, which means that
$$bar B(k_1, epsilon ), dots, bar B(k_{N^{rm ext}}, epsilon)$$
is an internal covering of size $N^{rm ext}$. Hence the smallest internal covering has size at most $N^{rm ext}$.
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
add a comment |
Suppose that $$bar B(x_1, epsilon / 2), dots, bar B(x_{N^{rm ext}}, epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i in { 1, dots, N^{rm ext}}$, there exists a $k_i in K$ that is contained in $bar B(x_i, epsilon / 2)$. (Otherwise $bar B(x_i, epsilon / 2)$ would be redundant, contradicting the minimality of size of the covering).
By the triangle inequality,
$$ bar B(x_i , epsilon / 2) subseteq bar B(k_i, epsilon), $$
for each $i$, which means that
$$bar B(k_1, epsilon ), dots, bar B(k_{N^{rm ext}}, epsilon)$$
is an internal covering of size $N^{rm ext}$. Hence the smallest internal covering has size at most $N^{rm ext}$.
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
Suppose that $$bar B(x_1, epsilon / 2), dots, bar B(x_{N^{rm ext}}, epsilon / 2)$$ is an external covering of $K$ of minimal size.
For each $i in { 1, dots, N^{rm ext}}$, there exists a $k_i in K$ that is contained in $bar B(x_i, epsilon / 2)$. (Otherwise $bar B(x_i, epsilon / 2)$ would be redundant, contradicting the minimality of size of the covering).
By the triangle inequality,
$$ bar B(x_i , epsilon / 2) subseteq bar B(k_i, epsilon), $$
for each $i$, which means that
$$bar B(k_1, epsilon ), dots, bar B(k_{N^{rm ext}}, epsilon)$$
is an internal covering of size $N^{rm ext}$. Hence the smallest internal covering has size at most $N^{rm ext}$.
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
edited Dec 28 '18 at 1:33
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
answered Dec 28 '18 at 1:01
Kenny Wong
18k21438
18k21438
add a comment |
add a comment |
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