Timeline for Axes of symmetry and symmetry group of the tangent cone to an open, connected, convex subset of the Euclidean space
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 14 at 1:02 | comment | added | Moishe Kohan | I see absolutely no way to assign any reasonable meaning to $R_l$ so that there your 3rd example has just three axes of symmetry. If $R_l$ is the unique isometric involution with fixed-point set $l$ then your 3rd example has continuum of axes of symmetry, all parallel to each other and contained in a common plane and no other axes of symmetry. | |
| Nov 14 at 0:51 | comment | added | Moishe Kohan | OK, what is $R_l$? | |
| Nov 14 at 0:40 | history | edited | Learning math | CC BY-SA 4.0 |
added 268 characters in body
|
| Nov 14 at 0:31 | history | edited | Learning math | CC BY-SA 4.0 |
added 268 characters in body
|
| Nov 14 at 0:29 | comment | added | Learning math | @MoisheKohan Thank you for your comment! Yes indeed, convex means indeed path connected, so connected, so that was redundant. Modified the question now! Also, an axis of symmetry is a line $l$ in the ambient Euclidean space with respect to which the reflection $R_l$ leaves the space invariant, i.e. if $x\in K \iff R_l(x) \in K.$ Added this bit into the question for further clarification. | |
| Nov 13 at 19:13 | comment | added | Moishe Kohan | You never defined an "axis of symmetry" of a cone. What does it mean? Also, do you realize that convex implies connected? | |
| Nov 13 at 16:42 | history | edited | Learning math | CC BY-SA 4.0 |
added 211 characters in body
|
| Nov 13 at 16:38 | comment | added | Learning math | @YCor You're right! Let me think how to rephrase the question! | |
| Nov 13 at 16:36 | comment | added | YCor | Yes, it's my point (just because the current phrasing takes some energy to define the tangent cone). | |
| Nov 13 at 16:28 | comment | added | Learning math | @YCor Apologies for any possible misunderstanding of your comment, but I read your statement as no need to refer to $K$ as convex. But I needed the convexity of $K$ to guarantee $T_0K$ is convex. I think you were asking me to rephrase the question for generalized convex cones, instead of tangent cones to convex open subsets. Is this correct? | |
| Nov 13 at 16:26 | comment | added | YCor | No it isn't, but this $K$ is not convex (while you're assuming $K$ convex in your post). | |
| Nov 13 at 16:24 | comment | added | Learning math | @YCor: if $K$ is just the union of nonnegative x and y-axes in $\mathbb{R}^2,$ is its tangent cone at the origin, $T_0K$ convex? | |
| Nov 13 at 16:20 | comment | added | YCor | The tangent cone is a convex cone based at $x$ and conversely for every convex cone $C$ based at $x$, the tangent cone is $C$ itself. So this is a question about symmetries of convex cones (no need to refer to convex sets and tangent cones). | |
| Nov 13 at 16:16 | history | asked | Learning math | CC BY-SA 4.0 |