What's the upper bound of the area of a $psi$-thickened perimeter of a curve?Looking to find the largest rectangle, by area, inside a polygonMust a curve of constant width be generated with an odd number of sides?Filling a big rectangle with smaller rectanglesA polygon compactness metric that filters out the noise of small concavities?Least area of maximal triangle inside convex $n$-gon.Does the perimeter of a polygon necessarily decrease if more edges are added to it, with the constraint of constant area?Which shapes could have the largest and lowest possible area?Does the interior angle for an optimized 2-field solution remain constant when going through N dimensions?proving that the area of a 2016 sided polygon is an even integerRatio of Perimeter^3 to the Area of an Isoceles Triangle.

Intuition of generalized eigenvector.

In Qur'an 7:161, why is "say the word of humility" translated in various ways?

The screen of my macbook suddenly broken down how can I do to recover

Fear of getting stuck on one programming language / technology that is not used in my country

Creature in Shazam mid-credits scene?

Where does the bonus feat in the cleric starting package come from?

Why electric field inside a cavity of a non-conducting sphere not zero?

Are the IPv6 address space and IPv4 address space completely disjoint?

Why does the Sun have different day lengths, but not the gas giants?

Has any country ever had 2 former presidents in jail simultaneously?

Why Shazam when there is already Superman?

Calculating Wattage for Resistor in High Frequency Application?

What should you do when eye contact makes your subordinate uncomfortable?

Can someone explain how this makes sense electrically?

How much character growth crosses the line into breaking the character

How to explain what's wrong with this application of the chain rule?

Problem with TransformedDistribution

What was the exact wording from Ivanhoe of this advice on how to free yourself from slavery?

What is Cash Advance APR?

Why did the Mercure fail?

Why did the EU agree to delay the Brexit deadline?

Delivering sarcasm

Should I outline or discovery write my stories?

Creepy dinosaur pc game identification



What's the upper bound of the area of a $psi$-thickened perimeter of a curve?


Looking to find the largest rectangle, by area, inside a polygonMust a curve of constant width be generated with an odd number of sides?Filling a big rectangle with smaller rectanglesA polygon compactness metric that filters out the noise of small concavities?Least area of maximal triangle inside convex $n$-gon.Does the perimeter of a polygon necessarily decrease if more edges are added to it, with the constraint of constant area?Which shapes could have the largest and lowest possible area?Does the interior angle for an optimized 2-field solution remain constant when going through N dimensions?proving that the area of a 2016 sided polygon is an even integerRatio of Perimeter^3 to the Area of an Isoceles Triangle.













0












$begingroup$


I'm reading through this paper and in the part Proof of Theorem 2 (page 4, at the bottom) I stumbled upon this problem.



Basically, we have a unit square partitioned into several districts $D_i$ with the area denoted by $|D_i|$ and perimeter by $|partial D_i|$. We take some constant $psi$ and make a "$psi$-thickened $partial D_i$", which I guess means constructing some kind of a belt around the perimeter, which has a constant width of $psi$ (but not constant distance from $delta D_i$). They then casually claim that this belt has an area bounded from above by
$$psileft|partial D_iright| + pi psi^2.$$



That makes intuitive sense when I imagine $D_i$ being a polygon, and I can even prove it for a $n$-sided polygon. The first part of the formula corresponds to rectangles with exactly $psi$ height drawn above every side of the polygon and the second part is an area of the circle which is a sum of all the circle-parts above the vertices.



The problem is, it can be any general curve that doesn't cross over itself. I'm not sure how to generalize this idea to curves.










share|cite|improve this question









$endgroup$











  • $begingroup$
    A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
    $endgroup$
    – Jean Marie
    Mar 15 at 18:24










  • $begingroup$
    Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
    $endgroup$
    – Sh4rP EYE
    Mar 15 at 18:30






  • 1




    $begingroup$
    Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
    $endgroup$
    – Jean Marie
    Mar 15 at 20:37











  • $begingroup$
    Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
    $endgroup$
    – Jean Marie
    Mar 15 at 20:52










  • $begingroup$
    Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
    $endgroup$
    – Sh4rP EYE
    Mar 16 at 9:26















0












$begingroup$


I'm reading through this paper and in the part Proof of Theorem 2 (page 4, at the bottom) I stumbled upon this problem.



Basically, we have a unit square partitioned into several districts $D_i$ with the area denoted by $|D_i|$ and perimeter by $|partial D_i|$. We take some constant $psi$ and make a "$psi$-thickened $partial D_i$", which I guess means constructing some kind of a belt around the perimeter, which has a constant width of $psi$ (but not constant distance from $delta D_i$). They then casually claim that this belt has an area bounded from above by
$$psileft|partial D_iright| + pi psi^2.$$



That makes intuitive sense when I imagine $D_i$ being a polygon, and I can even prove it for a $n$-sided polygon. The first part of the formula corresponds to rectangles with exactly $psi$ height drawn above every side of the polygon and the second part is an area of the circle which is a sum of all the circle-parts above the vertices.



The problem is, it can be any general curve that doesn't cross over itself. I'm not sure how to generalize this idea to curves.










share|cite|improve this question









$endgroup$











  • $begingroup$
    A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
    $endgroup$
    – Jean Marie
    Mar 15 at 18:24










  • $begingroup$
    Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
    $endgroup$
    – Sh4rP EYE
    Mar 15 at 18:30






  • 1




    $begingroup$
    Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
    $endgroup$
    – Jean Marie
    Mar 15 at 20:37











  • $begingroup$
    Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
    $endgroup$
    – Jean Marie
    Mar 15 at 20:52










  • $begingroup$
    Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
    $endgroup$
    – Sh4rP EYE
    Mar 16 at 9:26













0












0








0





$begingroup$


I'm reading through this paper and in the part Proof of Theorem 2 (page 4, at the bottom) I stumbled upon this problem.



Basically, we have a unit square partitioned into several districts $D_i$ with the area denoted by $|D_i|$ and perimeter by $|partial D_i|$. We take some constant $psi$ and make a "$psi$-thickened $partial D_i$", which I guess means constructing some kind of a belt around the perimeter, which has a constant width of $psi$ (but not constant distance from $delta D_i$). They then casually claim that this belt has an area bounded from above by
$$psileft|partial D_iright| + pi psi^2.$$



That makes intuitive sense when I imagine $D_i$ being a polygon, and I can even prove it for a $n$-sided polygon. The first part of the formula corresponds to rectangles with exactly $psi$ height drawn above every side of the polygon and the second part is an area of the circle which is a sum of all the circle-parts above the vertices.



The problem is, it can be any general curve that doesn't cross over itself. I'm not sure how to generalize this idea to curves.










share|cite|improve this question









$endgroup$




I'm reading through this paper and in the part Proof of Theorem 2 (page 4, at the bottom) I stumbled upon this problem.



Basically, we have a unit square partitioned into several districts $D_i$ with the area denoted by $|D_i|$ and perimeter by $|partial D_i|$. We take some constant $psi$ and make a "$psi$-thickened $partial D_i$", which I guess means constructing some kind of a belt around the perimeter, which has a constant width of $psi$ (but not constant distance from $delta D_i$). They then casually claim that this belt has an area bounded from above by
$$psileft|partial D_iright| + pi psi^2.$$



That makes intuitive sense when I imagine $D_i$ being a polygon, and I can even prove it for a $n$-sided polygon. The first part of the formula corresponds to rectangles with exactly $psi$ height drawn above every side of the polygon and the second part is an area of the circle which is a sum of all the circle-parts above the vertices.



The problem is, it can be any general curve that doesn't cross over itself. I'm not sure how to generalize this idea to curves.







geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Mar 15 at 18:00









Sh4rP EYESh4rP EYE

945




945











  • $begingroup$
    A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
    $endgroup$
    – Jean Marie
    Mar 15 at 18:24










  • $begingroup$
    Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
    $endgroup$
    – Sh4rP EYE
    Mar 15 at 18:30






  • 1




    $begingroup$
    Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
    $endgroup$
    – Jean Marie
    Mar 15 at 20:37











  • $begingroup$
    Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
    $endgroup$
    – Jean Marie
    Mar 15 at 20:52










  • $begingroup$
    Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
    $endgroup$
    – Sh4rP EYE
    Mar 16 at 9:26
















  • $begingroup$
    A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
    $endgroup$
    – Jean Marie
    Mar 15 at 18:24










  • $begingroup$
    Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
    $endgroup$
    – Sh4rP EYE
    Mar 15 at 18:30






  • 1




    $begingroup$
    Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
    $endgroup$
    – Jean Marie
    Mar 15 at 20:37











  • $begingroup$
    Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
    $endgroup$
    – Jean Marie
    Mar 15 at 20:52










  • $begingroup$
    Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
    $endgroup$
    – Sh4rP EYE
    Mar 16 at 9:26















$begingroup$
A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
$endgroup$
– Jean Marie
Mar 15 at 18:24




$begingroup$
A more current terminology is the dilated set. The formula you give works only for convex shapes with a regular boundary.
$endgroup$
– Jean Marie
Mar 15 at 18:24












$begingroup$
Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
$endgroup$
– Sh4rP EYE
Mar 15 at 18:30




$begingroup$
Great to hear that, you can't imagine how hard it is to google "constant-thickened perimeter". I'll look it up; however, they don't state or even mention any of those conditions.
$endgroup$
– Sh4rP EYE
Mar 15 at 18:30




1




1




$begingroup$
Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
$endgroup$
– Jean Marie
Mar 15 at 20:37





$begingroup$
Rectification : I had read too quickly : for an inequality, you do not need a convex set. It is for the equality : area of the dilated set by $psi $ = initial area + $psi times $ perimeter +$pi psi^2$ that you need a convex set.
$endgroup$
– Jean Marie
Mar 15 at 20:37













$begingroup$
Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
$endgroup$
– Jean Marie
Mar 15 at 20:52




$begingroup$
Other keywords for extension to 3D for example (that you hopefuly do not need) : Steiner's formula, Minkowski functionals
$endgroup$
– Jean Marie
Mar 15 at 20:52












$begingroup$
Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
$endgroup$
– Sh4rP EYE
Mar 16 at 9:26




$begingroup$
Thanks. I can't seem to find anything about dilated sets beyond 7th grade material, though. Could you please provide some further pointers, especially for the 2D version? Please excuse my inability to google properly.
$endgroup$
– Sh4rP EYE
Mar 16 at 9:26










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
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
);



);













draft saved

draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3149617%2fwhats-the-upper-bound-of-the-area-of-a-psi-thickened-perimeter-of-a-curve%23new-answer', 'question_page');

);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes















draft saved

draft discarded
















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • 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.




draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3149617%2fwhats-the-upper-bound-of-the-area-of-a-psi-thickened-perimeter-of-a-curve%23new-answer', 'question_page');

);

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







Popular posts from this blog

Solar Wings Breeze Design and development Specifications (Breeze) References Navigation menu1368-485X"Hang glider: Breeze (Solar Wings)"e

Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee

Method to test if a number is a perfect power? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Detecting perfect squares faster than by extracting square rooteffective way to get the integer sequence A181392 from oeisA rarely mentioned fact about perfect powersHow many numbers such $n$ are there that $n<100,lfloorsqrtn rfloor mid n$Check perfect squareness by modulo division against multiple basesFor what pair of integers $(a,b)$ is $3^a + 7^b$ a perfect square.Do there exist any positive integers $n$ such that $lfloore^nrfloor$ is a perfect power? What is the probability that one exists?finding perfect power factors of an integerProve that the sequence contains a perfect square for any natural number $m $ in the domain of $f$ .Counting Perfect Powers