Re: [math-fun] What is the simplex best at being best at?
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this. Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this. —Dan I wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.) (Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function F : C_n —> R there exists a global maximum and a global minimum on C_n. For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.) But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? ... ... -----
The wording of Dan’s question reminds me of a problem I’ve sometimes mused about: for what sort of n-athlon (whose n constituent events would not necessarily be sports-related in the narrow sense) would I be the world champion? How big would n have to be? (Has anyone written about the meta-game of designing games one would excel at?) Jim Propp On Wednesday, July 18, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this.
Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this.
—Dan
I wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.)
(Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function
F : C_n —> R
there exists a global maximum and a global minimum on C_n.
For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.)
But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? ... ... -----
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The email I sent last night was meant mostly as a joke, though I guess it was a little unclear what kind of reply I might be hoping for. In no sense did I intend a real math question. But I suspect that the meta-game of asking “Are there ten things I can do that nobody else can do all of? How about nine? How about eight?” (or a variant) has a name in pop (nerd) culture, and I was hoping to elicit it from one of you. (I suppose there are math questions lurking here, like “Given n IID samples from a d-dimensional Gaussian, how many hyperplanes on average does it take to isolate the first point from the other n-1?”, but I wasn’t asking then and I’m not asking them now.) Jim Propp On Wednesday, July 18, 2018, James Propp <jamespropp@gmail.com> wrote:
The wording of Dan’s question reminds me of a problem I’ve sometimes mused about: for what sort of n-athlon (whose n constituent events would not necessarily be sports-related in the narrow sense) would I be the world champion? How big would n have to be?
(Has anyone written about the meta-game of designing games one would excel at?)
Jim Propp
On Wednesday, July 18, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this.
Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this.
—Dan
I wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.)
(Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function
F : C_n —> R
there exists a global maximum and a global minimum on C_n.
For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.)
But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? ... ... -----
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I guess the Mind Sports Olympiad would be one possible good tournament to determine this: https://en.wikipedia.org/wiki/Mind_Sports_Olympiad (and I know you were not asking the question - but hey I take the bait, but the possible underlying math questions seems very similar to the technique called Isolation Forests for anomaly detection. Cheers, Seb On Thu, 19 Jul 2018 at 12:21, James Propp <jamespropp@gmail.com> wrote:
The email I sent last night was meant mostly as a joke, though I guess it was a little unclear what kind of reply I might be hoping for.
In no sense did I intend a real math question. But I suspect that the meta-game of asking “Are there ten things I can do that nobody else can do all of? How about nine? How about eight?” (or a variant) has a name in pop (nerd) culture, and I was hoping to elicit it from one of you.
(I suppose there are math questions lurking here, like “Given n IID samples from a d-dimensional Gaussian, how many hyperplanes on average does it take to isolate the first point from the other n-1?”, but I wasn’t asking then and I’m not asking them now.)
Jim Propp
On Wednesday, July 18, 2018, James Propp <jamespropp@gmail.com> wrote:
The wording of Dan’s question reminds me of a problem I’ve sometimes mused about: for what sort of n-athlon (whose n constituent events would not necessarily be sports-related in the narrow sense) would I be the world champion? How big would n have to be?
(Has anyone written about the meta-game of designing games one would excel at?)
Jim Propp
On Wednesday, July 18, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this.
Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this.
—Dan
I wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.)
(Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function
F : C_n —> R
there exists a global maximum and a global minimum on C_n.
For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.)
But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? ... ... -----
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I independently wondered the same question (your 'meta-game') when walking home several months ago. If you suppose every Thing is independent, then the proportion of people who are simultaneously better than you at the first n Things is expected to be a product: Z_1 x Z_2 x ... x Z_n where each Z_i is an iid uniform (0, 1) random variable. To be continued...
Sent: Thursday, July 19, 2018 at 11:20 AM From: "James Propp" <jamespropp@gmail.com> To: "Dan Asimov" <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] What is the simplex best at being best at?
The email I sent last night was meant mostly as a joke, though I guess it was a little unclear what kind of reply I might be hoping for.
In no sense did I intend a real math question. But I suspect that the meta-game of asking “Are there ten things I can do that nobody else can do all of? How about nine? How about eight?” (or a variant) has a name in pop (nerd) culture, and I was hoping to elicit it from one of you.
(I suppose there are math questions lurking here, like “Given n IID samples from a d-dimensional Gaussian, how many hyperplanes on average does it take to isolate the first point from the other n-1?”, but I wasn’t asking then and I’m not asking them now.)
Jim Propp
On Wednesday, July 18, 2018, James Propp <jamespropp@gmail.com> wrote:
The wording of Dan’s question reminds me of a problem I’ve sometimes mused about: for what sort of n-athlon (whose n constituent events would not necessarily be sports-related in the narrow sense) would I be the world champion? How big would n have to be?
(Has anyone written about the meta-game of designing games one would excel at?)
Jim Propp
On Wednesday, July 18, 2018, Dan Asimov <dasimov@earthlink.net> wrote:
I've had a few days to think about this. At this point, I'm pretty sure *what* I think is the number one distinction between the extremes of the regular simplex and the sphere, though I haven't yet convinced myself I know exactly *why* I believe this.
Meanwhile, if anyone else would like to express their opinion, I'd be curious to learn what others think about this.
—Dan
I wrote: ----- Let C_n be the space of convex bodies in R^n. (I.e., closed and bounded convex subsets of R^n that contain interior points.)
(Topologized with the Hausdorff metric, C_n is compact. Hence for any continuous function
F : C_n —> R
there exists a global maximum and a global minimum on C_n.
For many geometrically defined such F : C_n —> R, all spheres represent precisely the set of global maxima and all regular n-simplices represent precisely the set of global minima. (Or vice versa — same difference.)
But there is an embarrassment of options. Which F : C_n —> R best characterizes the gradient between the sphere and the simplex? ... ... -----
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participants (4)
-
Adam P. Goucher -
Dan Asimov -
James Propp -
Seb Perez-D