Let SH denote the homothety class generated by a convex set S ⊂ R: SH = {a + λS | a ∈ R, λ > 0}. We determine conditions for the Minkowski sum BH + CH or the Minkowski difference BH ∼ CH of homothety classes BH and CH generated by closed convex sets B,C ⊂ R to lie in a homothety class generated by a closed convex set (more generally, in the union of countably many homothety classes generated by...

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P .

Under the extra assumption that the boundary is of class C this was proven by S. Nakajima (= A. Matsumura) in 1926 Theorem 1 solves this problem. For convex bodies with C boundaries and positive curvature Nakajima’s result was generalized by Chakerian [?] in 1967 to “relative geometry” where the width and brightness are measured with with respect to some convex body K0 symmetric about the origi...

In this paper we study supervisory control of discrete event systems using state variables for representation and speci cation. The motivation is two fold: rstly, a state variable representation allows a concise characterization of systems with in nitely many states, and secondly, state variable speci cation allows characterization of qualitative properties of general nondeterministic systems. ...

Vacuum.-By creating a vacuum the process of decalcification can be characterized as follows: Insoluble calciurn-salts+acid o-soluble calciumsalts+CO2. The carbon dioxide, rapidly extracted, will disturb the balance of the chemical equilibrium, and, according to Le Chatelier, result in an acceleration of the reaction. Waerhaug (1949), however, points out that there will be a more intensive conta...

TheMinkowski sum of two setsA,B ∈ IR, denotedA⊕B, is defined as {a+ b | a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR using the convolution of the polygon boundaries. This method allows for faster computation of the sum of non-convex polygons in comparison to the widely-used methods for Minkowski-sum computation that d...