Affine buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to try to find analogs of results for symmetric spaces in the theory of buildings. In this paper we prove a version of Kostant's convexity theorem for thick non-discrete affine buildings. Kostant proves that the image of a certain orbit of a point $x$ in a symmetric space under a projection onto a maximal flat is the convex hull of the Weyl group orbit of $x$. We obtain the same result for a projection of a certain orbit of a point in an affine building to an apartment. The methods we use are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the final result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions onto apartments are distance diminishing.
For a function defined on a convex set in a Euclidean space , midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment . Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer lattice, we consider the analogous notion of discrete midpoint convexity, a discrete version of midpoint convexity where the value of the function at the (possibly noninteger) midpoint is replaced by …
In this paper, we study classes of discrete convex functions : submodular functions on modular semilattices and L- convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions , such as $k$ - submodular functions , skew-bisubmodular functions, L $^\natural$ - convex functions , tree submodular functions , and UJ- convex functions . We show that they are actually viewed as submodular/L- convex functions in our sense. We also prove a sharp iteration boun…
Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Applications to Contour Lines, Sparse Data and Inpainting
本文讨论了[K。Zhang，E. Crooks，A. Orlando，欧几里得空间中采样函数的近似和插值的补偿凸性方法：理论基础。SIAM Journal on Mathematical Analysis 48（2016）4126-4154]。我们应用我们的方法$(i)$表面重建开始从知识的许多水平集（或“轮廓线”）; $(ii)$分散的数据近似; $(iii)$图像修复。对于$(i)$和$(ii)$，我们的方法给出插值。对于有限集（分散的数据）的情况下，特别是，我们的近似提供了一个自然的三角剖分和分段仿射插值。显式计算的近似和修补结果的原型例子有限和紧凑的集合。我们还展示了我们的方法的应用程序的数值实验高密度盐和胡椒噪声降低图像处理中，图像修复和近似和插值的连续函数上采样的许多水平集和分散的点。
