Questions tagged [semialgebraic-geometry]

Semialgebraic geometry is the study of semialgebraic sets.

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Whether a Convex function is a semi-algebraic function, a convex set is a semi-algebraic set?

Definition: A semi-algebraic set is a subset described by equality and inequality of polynomials. For example: $\left\lbrace x \;|\; f(x)>0, \;g(x)=0, \ldots \right\rbrace$. Graph $(x,f(x))$ of a ...
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Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$ $$P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij},$$ I ...
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Duality gap in polynomial optimization problem Lasserre relaxation

Consider a polynomial optimization problem of the following type \begin{equation} \begin{array}{cl} \text{maximize} & p(\mathbf{x}) \\ \text{subject to} & \mathbf{x} \in K \\ \end{array} \...
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What geometry (if any) studies sets defined by infinitely many polynomial inequalities?

Question: Semi-algebraic geometry studies the solution sets of finitely many polynomial inequalities (in $\mathbb{R}^n$). What field (if any) could be considered the study of the solution sets of ...
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Area of semialgebraic set

Consider the following system of polynomial inequalities with variables $q_1,\ q_2$. \begin{align*} a_1 \leq q_1 + q_2 \leq a_2\\ b_1 \leq q_1^2 + q_2^2 \leq b_2\\ c_1 \leq q_1^3 + q_2^3 \leq c_2 \\ ...
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What is the volume of the $3$-dimensional elliptope?

My question Compute the following double integral analytically $$\int_{-1}^1 \int_{-1}^1 2 \sqrt{x^2 y^2 - x^2 - y^2 + 1} \,\, \mathrm{d} x \mathrm{d} y$$ Background The $3$-dimensional ...