Illustrating that universal optimality is stronger than sphere packingSphere packing in a sphereCritical Radius for Infinite Dimensional Sphere PackingExtensions of the Koebe–Andreev–Thurston theorem to sphere packing?maximal minimum distance in a sphere packingWhich term is better for the so called “sphere packing”?Computing the Volume of Closed 3-Manifolds and the Geometrization ConjectureTechniques for showing optimality of given packingTranslative packing constant strictly larger than lattice packing constantUnderstanding sphere packing in higher dimensionsSphere packing and kissing numbers in 3D
Illustrating that universal optimality is stronger than sphere packing
Sphere packing in a sphereCritical Radius for Infinite Dimensional Sphere PackingExtensions of the Koebe–Andreev–Thurston theorem to sphere packing?maximal minimum distance in a sphere packingWhich term is better for the so called “sphere packing”?Computing the Volume of Closed 3-Manifolds and the Geometrization ConjectureTechniques for showing optimality of given packingTranslative packing constant strictly larger than lattice packing constantUnderstanding sphere packing in higher dimensionsSphere packing and kissing numbers in 3D
$begingroup$
I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $mathbfR^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be universally optimal among point configurations, i.e., "they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians)."
I'm trying to appreciate the difference between sphere packing and universal optimality-- to this end, does anyone know of a simple example for which it is clear that the densest sphere packing is not universally optimal?
My initial intuition was that sphere packings would be necessarily universally optimal since you could imagine a spherical equipotential surface surrounding each particle, and that minimizing the energy of the configuration would amount to finding the optimal packing of these equipotential surfaces. Evidently this is false.
The paper with the aforementioned proof mentions that it was found in 3 dimensions that the conjectured optimal lattice solutions for potential functions of the form $r mapsto mathrme^- pi r^2$ are not optimal when nonlattice configurations are considered, but I am hoping for a more obvious illustration.
mg.metric-geometry global-optimization sphere-packing
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $mathbfR^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be universally optimal among point configurations, i.e., "they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians)."
I'm trying to appreciate the difference between sphere packing and universal optimality-- to this end, does anyone know of a simple example for which it is clear that the densest sphere packing is not universally optimal?
My initial intuition was that sphere packings would be necessarily universally optimal since you could imagine a spherical equipotential surface surrounding each particle, and that minimizing the energy of the configuration would amount to finding the optimal packing of these equipotential surfaces. Evidently this is false.
The paper with the aforementioned proof mentions that it was found in 3 dimensions that the conjectured optimal lattice solutions for potential functions of the form $r mapsto mathrme^- pi r^2$ are not optimal when nonlattice configurations are considered, but I am hoping for a more obvious illustration.
mg.metric-geometry global-optimization sphere-packing
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $mathbfR^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be universally optimal among point configurations, i.e., "they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians)."
I'm trying to appreciate the difference between sphere packing and universal optimality-- to this end, does anyone know of a simple example for which it is clear that the densest sphere packing is not universally optimal?
My initial intuition was that sphere packings would be necessarily universally optimal since you could imagine a spherical equipotential surface surrounding each particle, and that minimizing the energy of the configuration would amount to finding the optimal packing of these equipotential surfaces. Evidently this is false.
The paper with the aforementioned proof mentions that it was found in 3 dimensions that the conjectured optimal lattice solutions for potential functions of the form $r mapsto mathrme^- pi r^2$ are not optimal when nonlattice configurations are considered, but I am hoping for a more obvious illustration.
mg.metric-geometry global-optimization sphere-packing
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $mathbfR^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be universally optimal among point configurations, i.e., "they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians)."
I'm trying to appreciate the difference between sphere packing and universal optimality-- to this end, does anyone know of a simple example for which it is clear that the densest sphere packing is not universally optimal?
My initial intuition was that sphere packings would be necessarily universally optimal since you could imagine a spherical equipotential surface surrounding each particle, and that minimizing the energy of the configuration would amount to finding the optimal packing of these equipotential surfaces. Evidently this is false.
The paper with the aforementioned proof mentions that it was found in 3 dimensions that the conjectured optimal lattice solutions for potential functions of the form $r mapsto mathrme^- pi r^2$ are not optimal when nonlattice configurations are considered, but I am hoping for a more obvious illustration.
mg.metric-geometry global-optimization sphere-packing
mg.metric-geometry global-optimization sphere-packing
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
DiffycueDiffycue
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
DiffycueDiffycue
1233
asked 2 hours ago
DiffycueDiffycue
1233
1233
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Diffycue is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.
Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
$endgroup$
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Diffycue is a new contributor. Be nice, and check out our Code of Conduct.
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
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f331950%2fillustrating-that-universal-optimality-is-stronger-than-sphere-packing%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.
Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
$endgroup$
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
add a comment |
$begingroup$
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.
Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
$endgroup$
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
add a comment |
$begingroup$
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.
Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
$endgroup$
In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.
Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
answered 2 hours ago
Henry CohnHenry Cohn
14.2k25870
14.2k25870
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
add a comment |
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
$begingroup$
Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.)
$endgroup$
– Noam D. Elkies
1 hour ago
add a comment |
Diffycue is a new contributor. Be nice, and check out our Code of Conduct.
Diffycue is a new contributor. Be nice, and check out our Code of Conduct.
Diffycue is a new contributor. Be nice, and check out our Code of Conduct.
Diffycue is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f331950%2fillustrating-that-universal-optimality-is-stronger-than-sphere-packing%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
