# Tagged Questions

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### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex if its restriction to each line is. An affine ...
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### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $\lambda > 0$) and the ...
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### Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
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### Quermassintegrals as mean curvature integrals

It is well-known that the quermassintegrals $W_{i}$ of a convex body $K\subset \mathbb{R}^{n}$ can be written as a mean curvature integral  W_{i}(K)=n^{-1}\int_{\partial K}H_{i-1} ...
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Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail? Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ... 1answer 273 views ### Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and$d^2/2$-convexity Let$(M,g)$be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write$d_M(x,y)$for the (geodesic) distance between two points$x,y\in M$. I'm interested in the following class ... 3answers 442 views ### surfaces of constant centro-affine curvature It is well-known that every closed surface in$\mathbb R^3$having constant Gauss curvature is a round sphere. Inspired by this question, I'd like to ask whether a similar rigidity holds for ... 2answers 372 views ### Submanifolds lying on the boundary of a convex domain Let$M$be a submanifold of$\mathbb R^n$. Call$M$locally convex if locally$M$is contained in the boundary of a convex domain of$\mathbb R^n$. Is there any known condition that is equivalent to ... 2answers 325 views ### If$K$and$L\$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces. I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex"). Thanks!