Questions tagged [convex-geometry]

Use this tag when posting questions related to the concept of convexity and geometry. For example, for convex polygons.

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When does $A-B= B-A$ hold for bounded, closed, convex sets $A,B$?

When does $A-B=B-A$ hold for bounded, closed, convex sets $A,B\subset \mathbb{R}^n$? In 1D it is easy to see that this holds true if and only if the two intervals have the same center points. Is ...
1 vote
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Simple proof that John's theorem is sharp?

We denote $B^n_p \subset \mathbb{R}^n$ the unit ball $\{\|x\|_p \leq 1\} \subset \mathbb{R}^n$, $p \in [1, \infty]$. Let $\mathcal{K}^n$ denote the class of symmetric convex bodies in $\mathbb{R}^n$. ...
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Does the set where a convex function fails to be second derivative have some geometric properties? [closed]

I know it is of Hausdorff measure 0 by Alexsandrov's theorem. And the second derivative of a convex function can be viewed as a Radon measure. Does it have some other geometic properties?
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Normal Cones on a Circular Arc

Let $C = \{(x, y) \in \mathbb{R}^2: x^2 + y^2 \leq 1, y \geq 0, y \leq x\}.$ I wish to determine the normal cone $N_C(x_0, y_0)$, for any $(x_0, y_0)$ on the boundary of $C$. There are six cases to ...
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Covariances not below $\Sigma$

$\Sigma_0,\Sigma_1,\dots,\Sigma_K$ are real covariance matrices. I’m interested in the set of matrices $$\bigcap_{k=1}^K \left\{x: 0 \preceq x \preceq \Sigma_0, \ x\not\prec\Sigma_k\right\}.$$ I’m ...
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If $A$ has strictly positive reach, does the set $\{ x \in A \colon B(x,\epsilon) \subseteq A \}$ also?

Let $A \subseteq \mathbb{R}^n$ with $\text{reach}(A) > 0$ (see https://en.wikipedia.org/wiki/Reach_(mathematics) ). Define for any $\epsilon>0$, the "removal of $\epsilon$-thick boundary&...
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Polar Set of a Polyhedra is a Polytope

I am having trouble to verify Proposition 2 from the following MIT OCW document: Is there a way for us to see the equivalence of $C_1$ and $C_2$ simply through the definition? I do notice there was a ...
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Constrained Optimizer as Projection of Unconstrained Optimizer

Let $f:H\to \mathbb{R}$ be a once Fréchet differentiable strictly convex function on a separable Hilbert space and $C$ be a non-empty closed and convex body therein. Let $P_C:H\to C$ denote the ...
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1 vote