Re: [math-fun] Relative convexity
Maybe Jim could elaborate on how the desired concept relates to convexity. —Dan ----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y. In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p". ... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it. (I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!) Jim On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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In graph theory, there is the concept of the neighborhood N_G(X) of a set X of vertices in a graph G, namely, the set of vertices adjacent to some vertex in X. This is obviously generalizable to N_G(X,k), the set of vertices at distance k (or at most k) from some vertex in X. https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)#Neighbourhood_of_... See also https://mathworld.wolfram.com/search/?query=neighborhood+of+vertex&x=0&y=0 On Sat, Oct 24, 2020 at 2:42 PM James Propp <jamespropp@gmail.com> wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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Jim, is the following definition warm? Let S be some set of points, and let C be another set of points called the "core". For any point p in S, find the point c in C that is closest to p. Then define r(p) = p - c, the "position of p relative to C". (The subtraction is a vector subtraction.) Now construct the set S' = {r(p) | p in S}, the set of positions of all the points in p relative to C. Say that S is convex relative to C if S' is convex. For the motivating example, where C is a circle and S is a torus "around" C, S' will be a sphere. On Sat, Oct 24, 2020 at 3:19 PM W. Edwin Clark <wclark@mail.usf.edu> wrote:
In graph theory, there is the concept of the neighborhood N_G(X) of a set X of vertices in a graph G, namely, the set of vertices adjacent to some vertex in X. This is obviously generalizable to N_G(X,k), the set of vertices at distance k (or at most k) from some vertex in X.
https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)#Neighbourhood_of_...
See also https://mathworld.wolfram.com/search/?query=neighborhood+of+vertex&x=0&y=0
On Sat, Oct 24, 2020 at 2:42 PM James Propp <jamespropp@gmail.com> wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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I like the feature of Allan’s definition that makes an r-neighborhood of the core convex. Jim On Sat, Oct 24, 2020 at 5:12 PM Allan Wechsler <acwacw@gmail.com> wrote:
Jim, is the following definition warm? Let S be some set of points, and let C be another set of points called the "core". For any point p in S, find the point c in C that is closest to p. Then define r(p) = p - c, the "position of p relative to C". (The subtraction is a vector subtraction.)
Now construct the set S' = {r(p) | p in S}, the set of positions of all the points in p relative to C. Say that S is convex relative to C if S' is convex.
For the motivating example, where C is a circle and S is a torus "around" C, S' will be a sphere.
On Sat, Oct 24, 2020 at 3:19 PM W. Edwin Clark <wclark@mail.usf.edu> wrote:
In graph theory, there is the concept of the neighborhood N_G(X) of a set X of vertices in a graph G, namely, the set of vertices adjacent to some vertex in X. This is obviously generalizable to N_G(X,k), the set of vertices at distance k (or at most k) from some vertex in X.
https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)#Neighbourhood_of_...
See also
https://mathworld.wolfram.com/search/?query=neighborhood+of+vertex&x=0&y=0
On Sat, Oct 24, 2020 at 2:42 PM James Propp <jamespropp@gmail.com>
wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd
would
be
for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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In particular, an r-neighborhood of the core, relative to the core, is just a sphere of radius r. I think. That's what I was aiming for. On Sat, Oct 24, 2020 at 5:33 PM James Propp <jamespropp@gmail.com> wrote:
I like the feature of Allan’s definition that makes an r-neighborhood of the core convex.
Jim
On Sat, Oct 24, 2020 at 5:12 PM Allan Wechsler <acwacw@gmail.com> wrote:
Jim, is the following definition warm? Let S be some set of points, and let C be another set of points called the "core". For any point p in S, find the point c in C that is closest to p. Then define r(p) = p - c, the "position of p relative to C". (The subtraction is a vector subtraction.)
Now construct the set S' = {r(p) | p in S}, the set of positions of all the points in p relative to C. Say that S is convex relative to C if S' is convex.
For the motivating example, where C is a circle and S is a torus "around" C, S' will be a sphere.
On Sat, Oct 24, 2020 at 3:19 PM W. Edwin Clark <wclark@mail.usf.edu> wrote:
In graph theory, there is the concept of the neighborhood N_G(X) of a set X of vertices in a graph G, namely, the set of vertices adjacent to some vertex in X. This is obviously generalizable to N_G(X,k), the set of vertices at distance k (or at most k) from some vertex in X.
https://en.wikipedia.org/wiki/Neighbourhood_(graph_theory)#Neighbourhood_of_...
See also
https://mathworld.wolfram.com/search/?query=neighborhood+of+vertex&x=0&y=0
On Sat, Oct 24, 2020 at 2:42 PM James Propp <jamespropp@gmail.com>
wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize
the
definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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Here’s a definition that turns out not to be quite what I wanted but seems somewhat natural. Assume C is compact. Imagine you can teleport from any point in C to any other; then the ordinary metric d on R^3 gives rise to a “scrunched” quotient metric d/C on R^3 (to get from x to y you walk from x to a point nearest to x in C, teleport to a point nearest to y in C, and then walk to y), and a notion of a geodesic in the quotient metric. We could say that a set S that contains C is convex relative to C if there is a d/C geodesic that stays within S. This definition has the feature that if C is a circle and S is an r-neighborhood of C, then S is convex relative to C for small values of r but not for r close to the radius of C. I can’t decide how I feel about this. Jim Propp On Sat, Oct 24, 2020 at 2:41 PM James Propp <jamespropp@gmail.com> wrote:
It's more of a pre-theoretic feeling. I have a sense I'll recognize the definition I'm intuiting when I see it.
(I appreciate y'all being kinder to me than the MathOverflow crowd would be for my asking such a vague question!)
Jim
On Sat, Oct 24, 2020 at 1:43 PM Dan Asimov <dasimov@earthlink.net> wrote:
Maybe Jim could elaborate on how the desired concept relates to convexity.
—Dan
----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y.
In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p".
... ...
Jim Propp wrote: ----- Is there a notion of "relative convexity" that would make an ordinary torus consisting of points at small fixed distance from a large circle "convex relative to the circle"?
Thinking about different sorts of polyhedral tori people have come up with, I realize that part of what I want esthetically is some kind of relative convexity, but I don't know what it should mean! -----
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participants (4)
-
Allan Wechsler -
Dan Asimov -
James Propp -
W. Edwin Clark