Let be a real vector space and a symmetric bilinear form. In this post I will first show that Cauchy-Schwarz inequality is equivalent to being semi-definite. I will then show that for a non-negative positive-homogeneous function the triangle inequality, convexity, and quasi-convexity are all equivalent. Finally, I will show that if is such that . then is a semi-norm if and only if is semi-definite.
A bilinear form is called
for all .
A function is called
for all , , and .
Assume is indefinite. Then there exists such that and . It follows that does not hold; the left-hand side is non-negative and the right-hand side is negative. Therefore does not fulfill the Cauchy-Schwarz inequality.
Assume is semi-definite.
Let . Then
It follows that
Since is arbitrary, we have that . Therefore the Cauchy-Schwarz inequality holds.
Decompose as , where is the orthogonal projection of to , and is the rejection of from . By semi-definiteness, and ,
This is the Cauchy-Schwarz inequality.
Let be a convex function. Then
for all , and . Therefore is quasi-convex.
Let be a non-negative quasi-convex positive-homogeneous function. For , let . Now
Using positive-homogenuity and non-negativeness,
Therefore fulfills the triangle inequality.
Let be a positive-homogeneous function. Assume is convex. Then
for all . Therefore fulfills the triangle inequality. Assume fulfills the triangle inequality. Then
for all . Therefore f is convex.
Let be a symmetric bilinear form, and be such that .
Assume is indefinite. By indefiniteness, there exists such that and . Now one can solve the quadratic equation for . The solution is
The discriminant is always positive, since . Therefore, there are two points , with and , which lie on the same line as and . Either or is a convex combination of and . Without loss of generality, assume it is . If were convex, it would hold that . Since this is not the case, is not convex and the triangle inequality does not hold.
Assume is semi-definite. Then the Cauchy-Schwarz inequality holds and . Now,
Thus the triangle inequality holds.