A set is convex if, for any two points within the set, the line segment connecting them is also contained within the set. This property holds because the set is "bulging outwards" in all directions, allowing the line segment to remain entirely within the set. This property makes convex sets useful in optimization and mathematical analysis, as they have well-defined boundaries and allow for efficient algorithms to be applied. **

Non-automated files are documents or records that are not processed or managed by automated systems or software. These files are typically stored and managed manually, without the use of digital tools or technology. Non-automated files may include physical paper documents, handwritten notes, or any other type of information that is not stored or processed electronically. These files require manual handling and organization, and may be more susceptible to errors or inefficiencies compared to automated systems. **

Dorsal convex refers to a structure or surface that is curved outward or bulging towards the back or upper side of an organism. This term is commonly used in anatomy to describe the shape of certain body parts or bones. For example, the dorsal convexity of the spine refers to the natural outward curve of the spine towards the back of the body. **

The shape is concave. **

A convex pentagon is a five-sided polygon where all interior angles are less than 180 degrees and all vertices point outwards. This means that the shape does not have any indentations or concave angles. In a convex pentagon, no side crosses over another side, and the shape does not have any "dents" in its structure. **

A convex lens is a type of lens that is thicker in the middle and thinner at the edges. It is curved outward on both sides, causing light rays passing through it to converge at a single point called the focal point. Convex lenses are commonly used in magnifying glasses, cameras, and eyeglasses to focus light and create magnified images. They are also known as converging lenses because they cause light rays to converge. **

To determine if a function is convex, we need to check if the second derivative of the function is non-negative over its entire domain. If the second derivative is non-negative, then the function is convex. If the second derivative is negative at any point, then the function is not convex. **

Yes, there are coin-operated vending machines in Flensburg. These machines can be found in various locations such as train stations, shopping centers, and public buildings. They offer a variety of products including snacks, drinks, and even everyday essentials like toiletries and phone chargers. Customers can use coins to make purchases from these vending machines. **