We study two combinatorial parameters, which we denote by f(S) and h(S), associated with an arbitrary set S ⊆ Rd, where d ∈ N. In the nondegenerate situation, f(S) is the largest possible number of facets of a d-dimensional polyhedron L such that the interior of L is disjoint with S and L is inclusion-maximal with respect to this property. The parameter h(S) is the Helly number of the family of...

In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union. The space of pairs of convex bodies has been investigated in a number of papers [3], [8], [9], and [12]. This space has found an application in quasidifferential calculus (cf. [1], [5], [7], [10]). A quasidifferential is represented as a pair of convex bodies and it is essential to fin...

Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·, u, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it can be represented through a level conv...

We propose an online convex optimization algorithm (RESCALEDEXP) that achieves optimal regret in the unconstrained setting without prior knowledge of any bounds on the loss functions. We prove a lower bound showing an exponential separation between the regret of existing algorithms that require a known bound on the loss functions and any algorithm that does not require such knowledge. RESCALEDE...

In 1957, Chandler Davis proved that unitarily invariant convex functions on the space of hermitian matrices are precisely those which are convex and symmetrically invariant on the set of diagonal matrices. We give a simple perturbation theoretic proof of this result. (Davis’ argument was also very short, though based on completely different ideas). Consider an orthogonally invariant function f ...

We introduce a new method for computing the word length of an element of Thompson’s group F with respect to a “consecutive” generating set of the form Xn = {x0, x1, · · · , xn}, which is a subset of the standard infinite generating set for F . We use this method to show that (F, Xn) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.

The following conjecture of Fejes T6th is proved: The density of a lattice of convex bodies in [w” is at least i, provided that each connected component of the complementary set is bounded. The cases of equality are also determined. For any polytope P in [w” one can define a convexification P* (see Section 1) and it is proved that Vol P S Vol P*. It is also shown that if P and P’ are two convex...

A subset S of R is said to be polyhedral if it is the intersection of a finite number of closed halfspaces, i.e., if there exist J ∈ N and collections {y1, . . . , yJ} ⊂ R, {α1, . . . , αJ} ⊂ R such that S = ∩j=1{x ∈ R : y jx ≤ αj}. A function f : R → [−∞,+∞] is polyhedral if its epigraph epi f ⊂ R is a polyhedral set. Clearly, any polyhedral set is automatically convex and closed. Consequently...

We derive an a priori C2,α estimate for solutions of the fully non-linear elliptic equation F(D2u) = 0, provided the level set Σ = {M | F(M) = 0} satisfies: (a) Σ∩ {M | TrM = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal Fij to Σ is strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption on F in the...