# Tagged Questions

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### Convex compact set must have extreme points

I am reading a paper and there is such description as title. Why? I have an example: $(0,1)$. This is a convex set but not closed, so I cannot find an extreme point. However if convex and compact,...
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### How to give a formula of the perimeter of a $r$-neighborhood of a smooth set in $2D$?

Let $A$ be a simply connected open set in $\mathbb{R}^2$ with smooth boundary. Define $$A^r := \{x \in \mathbb{R}^2: d(x, A) \le r \},$$ where $d$ is the distance function. Let $P$ denote the ...
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### Continuous, midpoint (strictly) quasi-concave function is (strictly) quasi-concave?

It is known that Midpoint-Convex and Continuous Implies Convex. I am wondering can midpoint quasi-concavity and continuity implies quasi-concavity? If not, what conditions are required instead?
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### Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
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### Convexity of the weighted norm

We all know that $f(x)=\|x\|^2$, with $x\in\mathbb{R}^n$, is a strictly convex function of $x$. But know let's spicy up the problem. Let $v\in\mathbb{R}^n$ be a unit vector, i.e., $\|v\|=1$. We want ...
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### Is this constraint convex? Determinant of the Hessian is 0.

$a\leq e p_a D A (1-\Theta)$ $a,A$, and $\Theta$ are nonnegative decision variables and all others are positive parameters. Checking the Hessian tells me all of the leading principal minors are zero....
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### Gradient Descent and Scale of Data and Objective Function

One way to tune step size in gradient descent is via backtracking line search. backtracking line search (with parameters α ∈ (0, 1/2), β ∈ (0, 1)) starting at $t = 1$, repeat $t := \beta t$ ...
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### Is the root of a sum of squared differences convex?

Let $x \in \mathbb{R}^n$. Let there be a collection of functions $d_i = (x_j - x_k)^2$ (note that the subscripts $j$ and $k$ are fixed for each $d_i$, and there can be repeated use of subscripts on ...
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### Does the (strict) concavity of a function depends on the space in which we consider it?

For instance, $f(x)=\sqrt{x}$ is clearly strictly concave in $\mathbb{R}_+$ but if we consider that function in two dimensions, i.e. $f(x,y)=\sqrt{x}$ with $(x,y)\in\mathbb{R}^2_+$, it seems that it ...
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