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A convex set has the property that if you take any two points in the set and draw the line segment connecting those two points, that line segment lies entirely in the set.

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My textbook says that the figure on top represents a convex mirror and that the convex side of the spherical mirror faces the incident light. However, if I take two points $A$ and $P$ on this side, the line segment $AP$ does not lie entirely on this side. How, then, can the side of this mirror that faces the incident light be called the convex side?

Siddhartha
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    imagine a mirror as a surface of a solid body, for example of a ball, instead of a thin spherical layer. Then the solid is convex, and so we call the mirror. OTOH you need a niche in a reflecting body to geave a convex mirror. – CiaPan Jul 11 '18 at 16:54
  • @CiaPan Why would the outside of the ball be called the convex side (rather than the concave side), i.e., how does the outside of the ball relate to the mathematical definition of "convex"? – Siddhartha Jul 11 '18 at 17:26
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    This is not a mathematical, but a "semantical" problem. If you have a shell in the form of a half sphere its convex side is the outside. – Christian Blatter Jul 13 '18 at 18:54
  • I think that we shouldn’t be forcing mathematical terminology on the science of Optics. – Lubin Jul 15 '18 at 13:53
  • @Lubin OK, but there is no difference here. Both agree on what a convex mirror means. – Jens Jul 15 '18 at 14:31

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I agree that it is confusing that the boundary of a convex set, as seen from any point in that set, is actually concave, in the sense of curving in or hollowed inward. It might have been better if the set was called the "concave set".

In any case, if the boundary, as seen from any point within the set, is concave, it follows that the boundary, as seen from any point outside the set, must be convex.

Jens
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