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10 votes
1 answer
4k views

Uniform convergence of convex functions

It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform ov …
Roy Han's user avatar
  • 599
1 vote
1 answer
111 views

characterization of a certain closed convex cone

Consider $x_1,\cdots,x_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by $$\mathcal{K}(\underline{x}):=\{(\varphi(x_1),\cdots,\varphi(x_n)):\varphi \textrm{ convex on }\math …
Roy Han's user avatar
  • 599
2 votes
0 answers
142 views

error estimate of linear interpolation in high dimension

Consider convex functions $f,g$ on $[0,1]^d$. Let $x_1,\cdots,x_n$ be $n\geq d+1$ fixed point in $[0,1]^d$ that is equally 'distributed' in the sense that $$c_1\leq \frac{n\mathrm{vol}(K)}{\mathrm{car …
Roy Han's user avatar
  • 599
1 vote
1 answer
159 views

number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where $i_\alpha\in\{0,1,\ …
Roy Han's user avatar
  • 599
3 votes
1 answer
268 views

Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its …
Roy Han's user avatar
  • 599