User mini - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:06:01Z http://mathoverflow.net/feeds/user/12145 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61143/minimal-maximal-ellipsoids minimal maximal ellipsoids MINI 2011-04-09T15:02:17Z 2011-04-11T09:34:13Z <p>Suppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If $m\le\frac{\kappa}{s^3}(K)\le M$ for some positive numbers $m$ and $M$, does it mean there are ellipsoids $E_1$ and $E_2$ such that $E_1\subseteq K\subseteq E_2$ and $$\frac{\kappa}{s^3}({E_1})=M,~~~ \frac{\kappa}{s^3}(E_2)=m ~~~~?$$</p> http://mathoverflow.net/questions/59495/volume-of-the-projected-body volume of the projected body MINI 2011-03-24T22:01:56Z 2011-03-26T00:59:47Z <p>Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp P_1$ and $z_1\perp P_2$) and $K'$ is the projection of $K$ on $P_1\cap P_2$, what is the $Vol(K')$? We know the support function, and for simplicity let's suppose the body is symmetric and centered at the origin. If we just consider one hyperplane say, $P_0$, and want to compute the area of projection of $K$ on $P_0$ then the answer is $\frac{1}{2}\int_{\mathbb{S}^{n-1}}\frac{|\langle z,z_0\rangle|}{G}d\mu$ where $G$ is the Guass curvature of the boundary of $K$. I am looking for a solution of this type, possibly involving other symmetric functions of principle curvatures.</p> http://mathoverflow.net/questions/51704/lower-bound-on-the-curvature-of-the-curves-on-m Lower bound on the curvature of the curves on $M$ MINI 2011-01-10T21:38:49Z 2011-01-11T11:06:22Z <p>Let $M$ be an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, such that principal curvatures are bounded from below by a constant $\delta$. Is there any lower bound on the curvature of the curves on $M$? Curves should be intersection of a two plane and the manifold.</p> http://mathoverflow.net/questions/110579/good-diffeomorphisms Comment by MINI MINI 2012-10-24T21:09:51Z 2012-10-24T21:09:51Z I am terribly sorry. The range of f is positive real numbers. http://mathoverflow.net/questions/91231/affine-mean-curvature Comment by MINI MINI 2012-03-17T01:57:04Z 2012-03-17T01:57:04Z @Robert Bryant Via Gauss map. http://mathoverflow.net/questions/91231/affine-mean-curvature Comment by MINI MINI 2012-03-15T16:48:57Z 2012-03-15T16:48:57Z Using the affine normal vector: $\xi=:-h^{ki}\partial_i\mathcal{K}^{1/(n+2)}\partial_kX+\mathcal{K}^{1/(n+2)}\nu$, here $X:\mathcal{M}\to\mathcal{R}^{n+1}$ is an embedding and $\nu$ is the inward normal vector to $\mathcal{M}.$ http://mathoverflow.net/questions/80546/the-centro-affine-minkowski-problem/80555#80555 Comment by MINI MINI 2011-11-10T16:10:01Z 2011-11-10T16:10:01Z I asked her, but I wanted to be sure that I am not missing any known results. Thanks. http://mathoverflow.net/questions/73459/an-example-of-origin-sensitive-geometric-flow Comment by MINI MINI 2011-08-23T03:17:24Z 2011-08-23T03:17:24Z I know of that paper. In her paper she obtained those inequalities based on short time existence of the flow not long time existence. Any other example? http://mathoverflow.net/questions/72405/an-equation-in-differential-geometry Comment by MINI MINI 2011-08-08T22:53:00Z 2011-08-08T22:53:00Z $o$ mean composed and $K$ the Guass curvature is the determinant of $A[s]$ with respect to $\bar{g}.$ http://mathoverflow.net/questions/69566/normalizing-a-convex-curve Comment by MINI MINI 2011-07-09T14:21:03Z 2011-07-09T14:21:03Z if it is not embedded then it develops a self intersection. so there must be a piece of curve that has negative curvature but \gamma_{ss} is a convex curve, not strict convex though. lemma 7 the same paper as above( on affine plane curve evolution). http://mathoverflow.net/questions/69566/normalizing-a-convex-curve Comment by MINI MINI 2011-07-08T16:52:47Z 2011-07-08T16:52:47Z I think $\int_{\gamma}\mathcal{K}ds$ is twice the volume of the body enclosed by $\gamma_{ss}.$ Please kindly see equation (21) in <a href="http://www.sciencedirect.com/science/article/pii/S0022123684710044" rel="nofollow">sciencedirect.com/science/article/pii/&hellip;</a> in this paper $\mu$ stands for the affine curvature. http://mathoverflow.net/questions/69566/normalizing-a-convex-curve Comment by MINI MINI 2011-07-05T23:44:38Z 2011-07-05T23:44:38Z @Deane Yang Unfortunately that is the only version I have access to. http://mathoverflow.net/questions/69566/normalizing-a-convex-curve Comment by MINI MINI 2011-07-05T21:50:04Z 2011-07-05T21:50:04Z @Ben McKay i understood what you mean, i changed my notations. http://mathoverflow.net/questions/69566/normalizing-a-convex-curve Comment by MINI MINI 2011-07-05T21:44:56Z 2011-07-05T21:44:56Z @Will Jagy , for example, if $\gamma$ is an origin symmetric curve, then condition 21b is satisfied, thus an optimal length minimizing special linear transformation applied to $\gamma\in K_{\pi}$ should make both support function and curvature of $\gamma$ uniformly bounded with some absolute constants. that is what I understood, sounds weird to me. http://mathoverflow.net/questions/61143/minimal-maximal-ellipsoids/61276#61276 Comment by MINI MINI 2011-04-11T20:46:28Z 2011-04-11T20:46:28Z @Sergei Ivanov Superbe! http://mathoverflow.net/questions/61143/minimal-maximal-ellipsoids Comment by MINI MINI 2011-04-10T14:44:27Z 2011-04-10T14:44:27Z @Sergei Ivanov $\mathcal{K}/s^{n+1}$ is called centro-affine curvature. http://mathoverflow.net/questions/61143/minimal-maximal-ellipsoids Comment by MINI MINI 2011-04-10T14:43:20Z 2011-04-10T14:43:20Z @Sergei Ivanov I believe if $E\subseteq\mathbb{R}^n$ is an ellipsoid centered at the origin then $\mathcal{K}/s^{n+1}$ is constant, where $\mathcal{K}$ is the Gauss curvature of $\partial K$ the boundary of $K$. http://mathoverflow.net/questions/61143/minimal-maximal-ellipsoids Comment by MINI MINI 2011-04-09T18:33:33Z 2011-04-09T18:33:33Z @Sergei Ivanov edited.