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ChatGPT
Aug 19
1. All points on the line segment connecting any two points in the set are also in the set.
2. The set does not contain any holes, indentations, or protrusions.
3. The set lies entirely on one side of any line passing through two points in the set.
4. The set is closed under convex combinations, meaning that any convex combination of points in the set is also in the set.
5. The set is convex if and only if the line segment connecting any two points in the set lies entirely within the set.
2. The set does not contain any holes, indentations, or protrusions.
3. The set lies entirely on one side of any line passing through two points in the set.
4. The set is closed under convex combinations, meaning that any convex combination of points in the set is also in the set.
5. The set is convex if and only if the line segment connecting any two points in the set lies entirely within the set.
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Google Gemini AI
Aug 19
- Non-negative second derivative: If the second derivative of a function is non-negative at all points in its domain, the function is convex.
- A straight line between any two points on the graph lies above the graph: For any two points (x1, y1) and (x2, y2) on the graph of a convex function, the line segment connecting these two points lies above the graph.
- The function is increasing at an increasing rate: The first derivative of a convex function is increasing, which means that the rate of change of the function is increasing.
- The graph of a convex function is a smooth curve without any sharp corners or cusps.
- A convex function can be locally approximated by a linear function: At any point on the graph of a convex function, the tangent line to the graph is a good approximation to the function in a small neighborhood of that point.
- A straight line between any two points on the graph lies above the graph: For any two points (x1, y1) and (x2, y2) on the graph of a convex function, the line segment connecting these two points lies above the graph.
- The function is increasing at an increasing rate: The first derivative of a convex function is increasing, which means that the rate of change of the function is increasing.
- The graph of a convex function is a smooth curve without any sharp corners or cusps.
- A convex function can be locally approximated by a linear function: At any point on the graph of a convex function, the tangent line to the graph is a good approximation to the function in a small neighborhood of that point.
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