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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!

Q: How is a circle convex?

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convex

no

Yes.

None. Circles are convex shapes without straight sides and so there are no vertices (the correct plural of vertex).

a polygon is convex

Related questions

convex

no

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.

Yes.

Convex Polygon

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

All three are convex plane figures.

A circle, semicircle, segments or sectors of circle, ellipse, segments or sectors of ellipses, cardiods, closed convex wriggly shapes.

None. Circles are convex shapes without straight sides and so there are no vertices (the correct plural of vertex).

newtons ring is formed due to the consequtive circle of different radius of bright and dark in which the centre is dark

This depends on your definition. One could say that circles ({p in R^n : d(p,m) = r}) are concave, while (open) disks ({p in R^n : d(p,m) < r}) are convex.

a polygon is convex