a compact set with nonempty convex sections
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x in X$ and every coordinate $i=1,2,ldots,d$ denote by $x_-i := (x_j)_j neq i$.
Given a set $Y subseteq X$ and a vector $x_-i in [0,1]^d-1$ denote by $Y_x_-i := (x_i,x_-i) in Y colon x_i in [0,1]$ the $x_-i$-section of $Y$.
Let $Y subseteq X$ be a compact set that satisfies the following condition: for every $x in X$ and every coordinate $i=1,2,ldots,d$, the $x_-i$-section $Y_x_-i$ is nonempty and convex.
Is it true that the set $Y$ is contractible?
general-topology algebraic-topology convex-analysis convex-geometry
asked Sep 2 at 11:37
Eilon
112
add a comment |Â
up vote
0
down vote
favorite
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x in X$ and every coordinate $i=1,2,ldots,d$ denote by $x_-i := (x_j)_j neq i$.
Given a set $Y subseteq X$ and a vector $x_-i in [0,1]^d-1$ denote by $Y_x_-i := (x_i,x_-i) in Y colon x_i in [0,1]$ the $x_-i$-section of $Y$.
Let $Y subseteq X$ be a compact set that satisfies the following condition: for every $x in X$ and every coordinate $i=1,2,ldots,d$, the $x_-i$-section $Y_x_-i$ is nonempty and convex.
Is it true that the set $Y$ is contractible?
general-topology algebraic-topology convex-analysis convex-geometry
asked Sep 2 at 11:37
Eilon
112
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x in X$ and every coordinate $i=1,2,ldots,d$ denote by $x_-i := (x_j)_j neq i$.
Given a set $Y subseteq X$ and a vector $x_-i in [0,1]^d-1$ denote by $Y_x_-i := (x_i,x_-i) in Y colon x_i in [0,1]$ the $x_-i$-section of $Y$.
Let $Y subseteq X$ be a compact set that satisfies the following condition: for every $x in X$ and every coordinate $i=1,2,ldots,d$, the $x_-i$-section $Y_x_-i$ is nonempty and convex.
Is it true that the set $Y$ is contractible?
general-topology algebraic-topology convex-analysis convex-geometry
asked Sep 2 at 11:37
Eilon
112
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x in X$ and every coordinate $i=1,2,ldots,d$ denote by $x_-i := (x_j)_j neq i$.
Given a set $Y subseteq X$ and a vector $x_-i in [0,1]^d-1$ denote by $Y_x_-i := (x_i,x_-i) in Y colon x_i in [0,1]$ the $x_-i$-section of $Y$.
Let $Y subseteq X$ be a compact set that satisfies the following condition: for every $x in X$ and every coordinate $i=1,2,ldots,d$, the $x_-i$-section $Y_x_-i$ is nonempty and convex.
Is it true that the set $Y$ is contractible?
general-topology algebraic-topology convex-analysis convex-geometry
general-topology algebraic-topology convex-analysis convex-geometry
asked Sep 2 at 11:37
Eilon
112
asked Sep 2 at 11:37
Eilon
112
asked Sep 2 at 11:37
Eilon
112
asked Sep 2 at 11:37
Eilon
112
asked Sep 2 at 11:37
Eilon
112
112
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33
add a comment |Â
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2902628%2fa-compact-set-with-nonempty-convex-sections%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Do I understand correctly that each $Y_x_-i$ is a line segment? If so, I suggest to do define sections as follows. For $i = 1,ldots,d$ let $p_i : [0,1]^d to [0,1]^d-1$ denote the projection obtained by omitting the $i$-th coordinate. For $z in [0,1]^d-1$ let $$Y_i,z = Y cap p_i^-1(z) .$$
– Paul Frost
Sep 2 at 23:00
Indeed, Paul, your representation is equivalent to the one I provided.
– Eilon
Sep 3 at 5:33