Question

Do the following constitute convex sets in the 3 -space? (a) A doughnut (b) A bowling pin (c) A perfect marble

Short answer

The doughnut and bowling pin are not convex sets, but the perfect marble is a convex set.

Step-by-step solution

Understanding Convex Sets

A convex set is a set in which, for every pair of points within the set, the line segment joining them lies entirely within the set as well. In simpler terms, if you can pick any two points inside the set and draw a straight line between them without exiting the set, then the set is convex.

Examine a Doughnut

A doughnut shape, often referred to as a torus, has a hollow center. If you pick one point on the outer edge and another on the inner edge and draw a line between them, part of this line lies outside the doughnut structure. Therefore, the line does not entirely lie within the set, which means a doughnut is not a convex set.

Analyze a Bowling Pin

A bowling pin is a solid figure without any holes, so between any two points chosen within the structure, the line connecting them would typically lie within the material of the pin. However, due to its varying curvature and points along the surface, some tie lines (particularly those along the surface) may be external due to the bulging sections. Hence, a bowling pin is not entirely a convex set either.

Consider a Perfect Marble

A perfect marble, or sphere, is solid throughout without any internal hollow areas. Therefore, no matter where you select two points within it, the line connecting them will always remain within the sphere. Thus, a perfect marble is a convex set.

Key concepts

3-Space

The concept of "3-space" refers to the three-dimensional space that we live and perceive in, characterized by three coordinates: length, width, and height. In mathematics, understanding the properties of objects in 3-space is crucial. This space allows us to visualize objects with depth, unlike a flat 2D space. For instance, when examining a marble or a bowling pin in 3-space, we consider their entire structure in terms of thickness, height, and breadth.
3-space is often denoted by \(\mathbb{R}^3\), representing all points with three real number coordinates. This concept is foundational for studying geometry, physics, and engineering, as it helps in understanding how objects interact and occupy space in a realistic setting.

Geometry

Geometry is an area of mathematics that deals with shapes, sizes, and other properties of figures and spaces. It is essential in determining whether certain objects are convex. A set in geometry is considered convex if, for any two points within the set, the entire line segment connecting these points also lies wholly within the set.
Considering the geometric perspective, a doughnut, with its central hole, doesn't satisfy this convex property because certain lines can be drawn between two points within the doughnut that will exit the doughnut's boundary. Similarly, even though a bowling pin is a solid figure, its shape prevents it from being convex due to its contour, which can allow some surface points to form external lines. However, for a perfect marble, the spherical shape ensures that any two points within it can be connected entirely by a line that remains within the marble, making it a convex figure.

Mathematical Economics

Though mathematical economics might sound remote from the topic of convex sets, the concept plays a significant role in economic theories and models. In this field, convex sets often illustrate feasible regions or choices. A convex set ensures that any weighted average of choices represented by points within the set is also within the set, promoting the idea of mixture or diversification.

• Convexity in consumer preferences implies that blending two choices will not result in a less preferred outcome.
• In production, convex sets convey the idea that average outcomes or outputs from combining resources are feasible.

Thus, understanding convexity helps economists model efficient market scenarios and analyze optimal strategies for individuals and firms.

Solid Figures

Solid figures, also known as three-dimensional shapes, are fundamental in understanding concepts in 3-space and geometry. These figures include objects like spheres, cubes, and pyramids, among others. Solid figures have volume and surface area, making them a step beyond mere flat shapes.
When discussing convexity in solid figures, the object's entire volume and how every line segment within the figure lies are critical. For example, a sphere (like a perfect marble), with no hollows, satisfies the convex property because all internal lines between two points remain within the sphere. In contrast, a doughnut, though a solid figure, does not, due to its hole. Similarly, the varying contour of a bowling pin means not all connecting lines between points stay inside the figure, illustrating how not all solid figures can maintain convexity.