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 , then is a semi-norm if and only if is semi-definite.
A bilinear form is called
- symmetric, if ,
- positive-semi-definite, if ,
- negative-semi-definite, if ,
- semi-definite, if it is either positive-semi-definite or negative-semi-definite,
- indefinite, if it is not semi-definite, and
- fulfilling the Cauchy-Schwarz inequality, if
A function is called
- positive-homogeneous, if ,
- fulfilling the triangle inequality, if
- convex if , and
- quasi-convex, if
Cauchy-Schwarz inequality is equivalent to semi-definiteness
Assume the Cauchy-Schwarz inequality holds but that is indefinite. Then there exists such that and . Then does not hold since the left-hand side is non-negative and the right-hand side is negative. This is a contradiction. Therefore is semi-definite.
Assume is semi-definite. First, if , then the Cauchy-Schwarz inequality holds trivially. Assume . Decompose as , where is the orthogonal projection of to , and is the rejection of from . Then, by semi-definiteness, the Cauchy-Schwarz inequality states that
Again by semi-definiteness, and ,
Therefore the Cauchy-Schwarz inequality holds. QED.
Convexity implies quasi-convexity
Let be a convex function. Let , and . Now
Therefore is quasi-convex. QED.
Quasi-convexity + non-negativity + positive-homogenuity implies triangle inequality
Let be a non-negative quasi-convex positive-homogeneous function. For , let . Now
Using positive-homogenuity and non-negativeness,
Therefore fulfills the triangle inequality. QED.
For positive-homogeneous functions convexity and triangle inequality are equivalent
Let be a positive-homogeneous function. Assume is convex. Then Therefore fulfills the triangle inequality. Assume fulfills the triangle inequality. Let . Then . Therefore is convex. QED.
Semi-definiteness is equivalent to triangle inequality
Let be a symmetric bilinear form, and . Assume is indefinite. Then 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. QED.