Computing the Hausdorff Boundary Measure of Semialgebraic Sets

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.

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ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

International Journal of Mathematics ◽

10.1142/s0129167x12500310 ◽

2012 ◽

Vol 23 (04) ◽

pp. 1250031 ◽

Cited By ~ 8

Author(s):

JOSÉ F. FERNANDO ◽

J. M. GAMBOA

Keyword(s):

Integral Domain ◽

Hilbert Problem ◽

Semialgebraic Sets ◽

Semialgebraic Set ◽

Irreducible Components ◽

Real Geometry ◽

Nash Manifold ◽

Substitution Theorem ◽

Satisfactory Theory ◽

17Th Hilbert Problem

In this work we define a semialgebraic set S ⊂ ℝn to be irreducible if the noetherian ring [Formula: see text] of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring [Formula: see text]: Substitution Theorem, Positivstellensätze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.

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A measure for semialgebraic sets related to Boolean complexity

Mathematische Annalen ◽

10.1007/bf01446431 ◽

1989 ◽

Vol 283 (2) ◽

pp. 203-209 ◽

Cited By ~ 3

Author(s):

Gilbert Stengle

Keyword(s):

Semialgebraic Sets

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Characterization and description of basic semialgebraic sets

Algorithmic and Quantitative Real Algebraic Geometry - DIMACS Series in Discrete Mathematics and Theoretical Computer Science ◽

10.1090/dimacs/060/01 ◽

2003 ◽

pp. 1-12

Author(s):

C. Andradas

Keyword(s):

Semialgebraic Sets

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When is a Distribution of Signs Locally Completable?

Canadian Journal of Mathematics ◽

10.4153/cjm-1994-024-2 ◽

1994 ◽

Vol 46 (3) ◽

pp. 449-473 ◽

Cited By ~ 1

Author(s):

F. Acquistapace ◽

F. Broglia ◽

E. Fortuna

Keyword(s):

Algebraic Curve ◽

Algebraic Surface ◽

Regular Function ◽

Connected Components ◽

Semialgebraic Sets ◽

Sign Distribution

AbstractLet V be an irreducible nonsingular algebraic surface, Y ⊂ V be an algebraic curve and P a point of Y. Suppose a sign distribution is given locally in a neighbourhood of P on some connected components of V — Y. We give an algorithmic criterion to decide whether this sign distribution is induced by a regular function or not. As an application, this criterion enables one to decide whether two semialgebraic sets can be locally separated or not.

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