Question about smoothness of a Banach space.
$begingroup$
Define $delta: [0,2] to [0,1]$ defined by
$$delta_U(epsilon) = inf bigg{frac{1}{2}bigg(2-|u_1+u_2|bigg): u_1, u_2 in U^0, |u_1-u_2| geq epsilonbigg},$$ where $U^0$ is the boundary of the unit sphere in the Banach space $U$.
Then we say that $U$ is $p$-convex if $$delta_U(epsilon) geq Cepsilon^p$$ where $C$ is some constant.
Now, we define the concept of Bregman distance. Let $J_p$ is the duality mapping from $U$ to $U^*$ with the gauge function $t to t^{p-1}$. Then the functional $$Delta_p(u_1, u_2) = frac{1}{p}|u_1|^p-frac{1}{p}|u_2|^p-langle J_p(u_2), u_1-u_2rangle, qquad u_1, u_2 in U$$ is the Bregman distance of the convex functional $u to frac{1}{p}|u|^p$ at $u in U$.
Now we know that if $U$ is $p$-convex, then $$Delta_p(u_1, u_2) geq frac{K_p}{p}|u_1-u_2|^p$$
Can anyone provide an example of Banach space where $$p<K_p$$
Any help would be highly appreciated. Thanks in advance.
banach-spaces norm dual-spaces
edited Dec 4 '18 at 16:30
MOMO
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
$endgroup$
add a comment |
$begingroup$
Define $delta: [0,2] to [0,1]$ defined by
$$delta_U(epsilon) = inf bigg{frac{1}{2}bigg(2-|u_1+u_2|bigg): u_1, u_2 in U^0, |u_1-u_2| geq epsilonbigg},$$ where $U^0$ is the boundary of the unit sphere in the Banach space $U$.
Then we say that $U$ is $p$-convex if $$delta_U(epsilon) geq Cepsilon^p$$ where $C$ is some constant.
Now, we define the concept of Bregman distance. Let $J_p$ is the duality mapping from $U$ to $U^*$ with the gauge function $t to t^{p-1}$. Then the functional $$Delta_p(u_1, u_2) = frac{1}{p}|u_1|^p-frac{1}{p}|u_2|^p-langle J_p(u_2), u_1-u_2rangle, qquad u_1, u_2 in U$$ is the Bregman distance of the convex functional $u to frac{1}{p}|u|^p$ at $u in U$.
Now we know that if $U$ is $p$-convex, then $$Delta_p(u_1, u_2) geq frac{K_p}{p}|u_1-u_2|^p$$
Can anyone provide an example of Banach space where $$p<K_p$$
Any help would be highly appreciated. Thanks in advance.
banach-spaces norm dual-spaces
edited Dec 4 '18 at 16:30
MOMO
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
$endgroup$
add a comment |
$begingroup$
Define $delta: [0,2] to [0,1]$ defined by
$$delta_U(epsilon) = inf bigg{frac{1}{2}bigg(2-|u_1+u_2|bigg): u_1, u_2 in U^0, |u_1-u_2| geq epsilonbigg},$$ where $U^0$ is the boundary of the unit sphere in the Banach space $U$.
Then we say that $U$ is $p$-convex if $$delta_U(epsilon) geq Cepsilon^p$$ where $C$ is some constant.
Now, we define the concept of Bregman distance. Let $J_p$ is the duality mapping from $U$ to $U^*$ with the gauge function $t to t^{p-1}$. Then the functional $$Delta_p(u_1, u_2) = frac{1}{p}|u_1|^p-frac{1}{p}|u_2|^p-langle J_p(u_2), u_1-u_2rangle, qquad u_1, u_2 in U$$ is the Bregman distance of the convex functional $u to frac{1}{p}|u|^p$ at $u in U$.
Now we know that if $U$ is $p$-convex, then $$Delta_p(u_1, u_2) geq frac{K_p}{p}|u_1-u_2|^p$$
Can anyone provide an example of Banach space where $$p<K_p$$
Any help would be highly appreciated. Thanks in advance.
banach-spaces norm dual-spaces
edited Dec 4 '18 at 16:30
MOMO
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
$endgroup$
Define $delta: [0,2] to [0,1]$ defined by
$$delta_U(epsilon) = inf bigg{frac{1}{2}bigg(2-|u_1+u_2|bigg): u_1, u_2 in U^0, |u_1-u_2| geq epsilonbigg},$$ where $U^0$ is the boundary of the unit sphere in the Banach space $U$.
Then we say that $U$ is $p$-convex if $$delta_U(epsilon) geq Cepsilon^p$$ where $C$ is some constant.
Now, we define the concept of Bregman distance. Let $J_p$ is the duality mapping from $U$ to $U^*$ with the gauge function $t to t^{p-1}$. Then the functional $$Delta_p(u_1, u_2) = frac{1}{p}|u_1|^p-frac{1}{p}|u_2|^p-langle J_p(u_2), u_1-u_2rangle, qquad u_1, u_2 in U$$ is the Bregman distance of the convex functional $u to frac{1}{p}|u|^p$ at $u in U$.
Now we know that if $U$ is $p$-convex, then $$Delta_p(u_1, u_2) geq frac{K_p}{p}|u_1-u_2|^p$$
Can anyone provide an example of Banach space where $$p<K_p$$
Any help would be highly appreciated. Thanks in advance.
banach-spaces norm dual-spaces
banach-spaces norm dual-spaces
edited Dec 4 '18 at 16:30
MOMO
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
edited Dec 4 '18 at 16:30
MOMO
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
edited Dec 4 '18 at 16:30
MOMO
717312
edited Dec 4 '18 at 16:30
MOMO
717312
edited Dec 4 '18 at 16:30
MOMO
717312
717312
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
asked Dec 4 '18 at 14:00
Mittal GMittal G
1,252516
1,252516
add a comment |
add a comment |
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