# Tagged Questions

148 views

### The epigraph of a semi-convex function has positive reach

I've been trying to prove the following theorem for several hours with no result so far. Claim. Let $f:\mathbb{R} \to \mathbb{R}$ be a semi-convex function, i.e. there exists a constant $C > 0$ ...
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### Continuity of the real Monge Ampère operator on convex functions

Let $E\subset\mathbb{R}^n$ be a convex set, $u:E\to\mathbb{R}$ a convex function and $B\subset E$ a Borel set. We define the (multivalued) gradient \begin{array}{rccl} \nabla [u]:& \mathbb{R}^n ...
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### Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,  \mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
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### Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \frac{1}{b-a} \int_a^b ...
While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where $\lambda_{\text{min}}$ ...