Q: Is a circle a convex set?

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Yes.

convex

yes

no, because it should be a segment .

I'm not sure whether you are asking what convex means, or to prove that all circles are convex. A shape is convex if all pairs of points within it (i.e., the interior, not just the boundary) have all points between them within the shape. That is, if you draw the straight line segment between 2 points on or within a circle, all points of that line segment are within the circle. To prove that, you could use Euclid's axioms of geometry, or analytic geometry, which deals with the coordinates of points and the equations they satisfy. I will leave the proof as an exercise, since it is rather involved (but not necessarily advanced) and I don't know if that is what you are asking for anyway!

Related questions

Yes.

convex

the union of two convex sets need not be a convex set.

No, the region enclosed by a circle is not considered convex because it contains points within the circle that do not lie on the boundary of the circle. In convex regions, any line segment connecting two points inside the region will also lie completely inside the region.

Convex Polygon

Convex refers to a shape or surface that curves outward like the exterior of a circle. In mathematics, it describes a set where any line segment connecting two points within the set lies completely within the set. Convexity is often used in optimization and geometry to simplify problem-solving.

yes

A closed, convex, plane (2-dimensional) shape.

All three are convex plane figures.

The answer depends on how it is halved. If the plane is divided in two by a step graph (a zig-zag line) then it will not be a convex set.

A convex polygon is one with no reflex angles (angles that measure more than 180 degrees when viewed from inside the polygon). More generally a convex set is on where a straight line between any two points in the set lies completely within the set.

no, because it should be a segment .